4.5. Harmonic-analysis method: the quadratic-form network🔗

Section 7 of the article is the one architecture whose parameter space is not a copy of the data space, and it is where the formalization has the most to say. A quadratic form in the data followed by an arbitrary activation is jointly invariant under the affine group acting linearly on the coefficients, so the abstract reconstruction theorem applies once a balanced parameter measure exists; but the measure has to be built, the boundedness appendix cannot be used as written, and the route that replaces it is new analysis. This chapter follows that arc: the algebra, the two parameter measures, the endpoint from bounded intertwiners, the two square-integrability routes and the compactness obstruction that closes them, and then the intermediate coefficient space through which the analysis and the synthesis are bounded separately.

The algebra and the parameter measures

Lemma4.5.1

Statement uses 2

Statement dependency previews

Preview

Lemma 4.2.4

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

Used by 3

Reverse dependency previews

Preview

Theorem 4.5.2

Loading preview

Reverse dependency preview content is loaded from the rendered-fragment cache.

✓L∃∀N

Section 7. A quadratic form in the data followed by an arbitrary activation, \sigma(x^\top A x + x^\top b + c), with the affine group acting on the data by x\mapsto Lx+t and on the parameters by (A,b,c)\mapsto(L^{-\top}AL^{-1},\ L^{-\top}b-2L^{-\top}AL^{-1}t,\ c+t^\top L^{-\top}AL^{-1}t-t^\top L^{-\top}b).

Lean code for Lemma4.5.1●27 declarations

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.symmetricSubmodule.{u_2} (E : Type u_2)
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] : Submodule ℝ (E →L[ℝ] E)

def LeanRidgelet.symmetricSubmodule.{u_2}
  (E : Type u_2) [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  Submodule ℝ (E →L[ℝ] E)

Implementation after :=

:= {A | IsSelfAdjoint A}
  add_mem' ha hb := IsSelfAdjoint.add ha hb
  zero_mem' := IsSelfAdjoint.zero _
  smul_mem' r _ ha := (IsSelfAdjoint.all r).smul ha

The self-adjoint continuous endomorphisms, as a real subspace.

abbrevdefined in LeanRidgelet/HA/Quadratic.lean

complete

abbrev LeanRidgelet.QuadraticSymmetric.{u_2} (E : Type u_2)
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] : Type u_2

abbrev LeanRidgelet.QuadraticSymmetric.{u_2}
  (E : Type u_2) [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] : Type u_2

Implementation after :=

:= ↥(symmetricSubmodule E)

The symmetric coefficient of a quadratic-form feature.

abbrevdefined in LeanRidgelet/HA/Quadratic.lean

complete

abbrev LeanRidgelet.QuadraticParameter.{u_2} (E : Type u_2)
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] : Type u_2

abbrev LeanRidgelet.QuadraticParameter.{u_2}
  (E : Type u_2) [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] : Type u_2

Implementation after :=

:= QuadraticSymmetric E × E × ℝ

The parameter space `(A, b, c)` of a quadratic-form feature.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticCongrEndo.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (M : E ≃ₗ[ℝ] E) : E →L[ℝ] E

def LeanRidgelet.quadraticCongrEndo.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (M : E ≃ₗ[ℝ] E) : E →L[ℝ] E

Implementation after :=

:=
  LinearMap.toContinuousLinearMap (M : E →ₗ[ℝ] E)

A linear equivalence of a finite-dimensional space, viewed as a continuous endomorphism.  In
finite dimensions every linear map is continuous, so this is only a change of bundling.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticEval.{u_1} {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E] (t : E) : (E →L[ℝ] E) →ₗ[ℝ] E

def LeanRidgelet.quadraticEval.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E] (t : E) :
  (E →L[ℝ] E) →ₗ[ℝ] E

Implementation after :=

:= A t
  map_add' _ _ := rfl
  map_smul' _ _ := rfl

Evaluation of a continuous endomorphism at a fixed vector, as a linear map.

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.isSelfAdjoint_congr.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (M : E →L[ℝ] E) {A : E →L[ℝ] E} (hA : IsSelfAdjoint A) :
  IsSelfAdjoint (star M * A * M)

theorem LeanRidgelet.isSelfAdjoint_congr.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (M : E →L[ℝ] E)
  {A : E →L[ℝ] E} (hA : IsSelfAdjoint A) :
  IsSelfAdjoint (star M * A * M)

Congruence `A ↦ Mᵀ A M` preserves self-adjointness.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticCongrMap.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (M : E →L[ℝ] E) :
  LeanRidgelet.QuadraticSymmetric E →ₗ[ℝ]
    LeanRidgelet.QuadraticSymmetric E

def LeanRidgelet.quadraticCongrMap.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (M : E →L[ℝ] E) :
  LeanRidgelet.QuadraticSymmetric E →ₗ[ℝ]
    LeanRidgelet.QuadraticSymmetric E

Implementation after :=

:= ⟨star M * (A : E →L[ℝ] E) * M, isSelfAdjoint_congr M A.2⟩
  map_add' A B := by
    refine Subtype.ext ?_
    change star M * ((A : E →L[ℝ] E) + B) * M = star M * (A : E →L[ℝ] E) * M + star M * B * M
    rw [mul_add, add_mul]
  map_smul' r A := by
    refine Subtype.ext ?_
    change star M * (r • (A : E →L[ℝ] E)) * M = r • (star M * (A : E →L[ℝ] E) * M)
    rw [mul_smul_comm, smul_mul_assoc]

Congruence `A ↦ Mᵀ A M` on symmetric coefficients.

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.quadraticCongrMap_coe.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (M : E →L[ℝ] E) (A : LeanRidgelet.QuadraticSymmetric E) :
  ↑((LeanRidgelet.quadraticCongrMap M) A) = star M * ↑A * M

theorem LeanRidgelet.quadraticCongrMap_coe.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (M : E →L[ℝ] E)
  (A :
    LeanRidgelet.QuadraticSymmetric E) :
  ↑((LeanRidgelet.quadraticCongrMap M)
        A) =
    star M * ↑A * M

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticCongr.{u_1} {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] (M : E ≃ₗ[ℝ] E) :
  LeanRidgelet.QuadraticSymmetric E ≃ₗ[ℝ]
    LeanRidgelet.QuadraticSymmetric E

def LeanRidgelet.quadraticCongr.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (M : E ≃ₗ[ℝ] E) :
  LeanRidgelet.QuadraticSymmetric E ≃ₗ[ℝ]
    LeanRidgelet.QuadraticSymmetric E

Implementation after :=

:=
  LinearEquiv.ofLinear (quadraticCongrMap (quadraticCongrEndo M))
    (quadraticCongrMap (quadraticCongrEndo M.symm))
    (by
      have hinv : quadraticCongrEndo M.symm * quadraticCongrEndo M = 1 :=
        ContinuousLinearMap.ext fun x ↦ by simp [quadraticCongrEndo]
      refine LinearMap.ext fun A ↦ Subtype.ext ?_
      set N := quadraticCongrEndo M
      set N' := quadraticCongrEndo M.symm
      change star N * (star N' * (A : E →L[ℝ] E) * N') * N = (A : E →L[ℝ] E)
      calc star N * (star N' * (A : E →L[ℝ] E) * N') * N
          = star (N' * N) * (A : E →L[ℝ] E) * (N' * N) := by
            rw [star_mul]; simp only [mul_assoc]
        _ = (A : E →L[ℝ] E) := by rw [hinv]; simp)
    (by
      have hinv : quadraticCongrEndo M * quadraticCongrEndo M.symm = 1 :=
        ContinuousLinearMap.ext fun x ↦ by simp [quadraticCongrEndo]
      refine LinearMap.ext fun A ↦ Subtype.ext ?_
      set N := quadraticCongrEndo M
      set N' := quadraticCongrEndo M.symm
      change star N' * (star N * (A : E →L[ℝ] E) * N) * N' = (A : E →L[ℝ] E)
      calc star N' * (star N * (A : E →L[ℝ] E) * N) * N'
          = star (N * N') * (A : E →L[ℝ] E) * (N * N') := by
            rw [star_mul]; simp only [mul_assoc]
        _ = (A : E →L[ℝ] E) := by rw [hinv]; simp)

Congruence by an invertible map is invertible, with inverse the congruence by the inverse.

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.quadraticCongr_coe.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (M : E ≃ₗ[ℝ] E) (A : LeanRidgelet.QuadraticSymmetric E) :
  ↑((LeanRidgelet.quadraticCongr M) A) =
    star (LeanRidgelet.quadraticCongrEndo M) * ↑A *
      LeanRidgelet.quadraticCongrEndo M

theorem LeanRidgelet.quadraticCongr_coe.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (M : E ≃ₗ[ℝ] E)
  (A :
    LeanRidgelet.QuadraticSymmetric E) :
  ↑((LeanRidgelet.quadraticCongr M) A) =
    star
          (LeanRidgelet.quadraticCongrEndo
            M) *
        ↑A *
      LeanRidgelet.quadraticCongrEndo M

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.inner_quadraticCongr.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (M : E ≃ₗ[ℝ] E) (A : LeanRidgelet.QuadraticSymmetric E) (u v : E) :
  inner ℝ u (↑((LeanRidgelet.quadraticCongr M) A) v) =
    inner ℝ (M u) (↑A (M v))

theorem LeanRidgelet.inner_quadraticCongr.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (M : E ≃ₗ[ℝ] E)
  (A : LeanRidgelet.QuadraticSymmetric E)
  (u v : E) :
  inner ℝ u
      (↑((LeanRidgelet.quadraticCongr M)
            A)
        v) =
    inner ℝ (M u) (↑A (M v))

Congruence in an inner product: `⟪u, Mᵀ A M v⟫ = ⟪M u, A (M v)⟫`.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticArgument.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (x : E) (ξ : LeanRidgelet.QuadraticParameter E) : ℝ

def LeanRidgelet.quadraticArgument.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (x : E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  ℝ

Implementation after :=

:=
  ⟪x, (ξ.1 : E →L[ℝ] E) x⟫_ℝ + ⟪x, ξ.2.1⟫_ℝ + ξ.2.2

The scalar functional `xᵀ A x + xᵀ b + c` computed by a quadratic-form feature.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticFeature.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (σ : ℝ → ℝ) (x : E) (ξ : LeanRidgelet.QuadraticParameter E) : ℝ

def LeanRidgelet.quadraticFeature.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (σ : ℝ → ℝ)
  (x : E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  ℝ

Implementation after :=

:=
  σ (quadraticArgument x ξ)

The quadratic-form feature `σ(xᵀ A x + xᵀ b + c)` of Section 7.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticShearVector.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : LeanRidgelet.QuadraticSymmetric E →ₗ[ℝ] E

def LeanRidgelet.quadraticShearVector.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.QuadraticSymmetric E →ₗ[ℝ]
    E

Implementation after :=

:=
  (quadraticEval (g 0)).comp
    ((symmetricSubmodule E).subtype.comp (quadraticCongr g.linear.symm).toLinearMap)

The transformed symmetric coefficient applied to the translation of `g`.

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.quadraticShearVector_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (A : LeanRidgelet.QuadraticSymmetric E) :
  (LeanRidgelet.quadraticShearVector g) A =
    ↑((LeanRidgelet.quadraticCongr g.linear.symm) A) (g 0)

theorem LeanRidgelet.quadraticShearVector_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (A :
    LeanRidgelet.QuadraticSymmetric E) :
  (LeanRidgelet.quadraticShearVector g)
      A =
    ↑((LeanRidgelet.quadraticCongr
            g.linear.symm)
          A)
      (g 0)

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticLinearShear.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : E →ₗ[ℝ] ℝ

def LeanRidgelet.quadraticLinearShear.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : E →ₗ[ℝ] ℝ

Implementation after :=

:=
  -(innerSL ℝ (g 0)).toLinearMap.comp (g.linear.symm.adjoint : E →ₗ[ℝ] E)

The translation-dependent block acting on the linear coefficient of a quadratic parameter.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticLinearEquiv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : (E × ℝ) ≃ₗ[ℝ] E × ℝ

def LeanRidgelet.quadraticLinearEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : (E × ℝ) ≃ₗ[ℝ] E × ℝ

Implementation after :=

:=
  g.linear.symm.adjoint.skewProd (LinearEquiv.refl ℝ ℝ) (quadraticLinearShear g)

The action on the linear and constant coefficients alone.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticSymmetricShear.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : LeanRidgelet.QuadraticSymmetric E →ₗ[ℝ] E × ℝ

def LeanRidgelet.quadraticSymmetricShear.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.QuadraticSymmetric E →ₗ[ℝ]
    E × ℝ

Implementation after :=

:=
  ((-2 : ℝ) • quadraticShearVector g).prod
    ((innerSL ℝ (g 0)).toLinearMap.comp (quadraticShearVector g))

The translation-dependent block below the symmetric coefficient.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticParameterLinearEquiv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.QuadraticParameter E ≃ₗ[ℝ]
    LeanRidgelet.QuadraticParameter E

def LeanRidgelet.quadraticParameterLinearEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.QuadraticParameter E ≃ₗ[ℝ]
    LeanRidgelet.QuadraticParameter E

Implementation after :=

:=
  (quadraticCongr g.linear.symm).skewProd (quadraticLinearEquiv g) (quadraticSymmetricShear g)

The parameter action of the affine group on quadratic parameters.  If `g x = L x + t`, it
sends `(A, b, c)` to `(L⁻ᵀ A L⁻¹, L⁻ᵀ b - 2 L⁻ᵀ A L⁻¹ t, c + tᵀ L⁻ᵀ A L⁻¹ t - tᵀ L⁻ᵀ b)`.

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.quadraticParameterLinearEquiv_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (ξ : LeanRidgelet.QuadraticParameter E) :
  (LeanRidgelet.quadraticParameterLinearEquiv g) ξ =
    ((LeanRidgelet.quadraticCongr g.linear.symm) ξ.1,
      g.linear.symm.adjoint ξ.2.1 +
        -2 • (LeanRidgelet.quadraticShearVector g) ξ.1,
      ξ.2.2 + (LeanRidgelet.quadraticLinearShear g) ξ.2.1 +
        inner ℝ (g 0) ((LeanRidgelet.quadraticShearVector g) ξ.1))

theorem LeanRidgelet.quadraticParameterLinearEquiv_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  (LeanRidgelet.quadraticParameterLinearEquiv
        g)
      ξ =
    ((LeanRidgelet.quadraticCongr
          g.linear.symm)
        ξ.1,
      g.linear.symm.adjoint ξ.2.1 +
        -2 •
          (LeanRidgelet.quadraticShearVector
              g)
            ξ.1,
      ξ.2.2 +
          (LeanRidgelet.quadraticLinearShear
              g)
            ξ.2.1 +
        inner ℝ (g 0)
          ((LeanRidgelet.quadraticShearVector
              g)
            ξ.1))

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.quadraticArgument_invariant.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (x : E) (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticArgument (g x)
      ((LeanRidgelet.quadraticParameterLinearEquiv g) ξ) =
    LeanRidgelet.quadraticArgument x ξ

theorem LeanRidgelet.quadraticArgument_invariant.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (x : E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticArgument (g x)
      ((LeanRidgelet.quadraticParameterLinearEquiv
          g)
        ξ) =
    LeanRidgelet.quadraticArgument x ξ

The quadratic functional is invariant under the joint affine data/parameter action.  This is
the computation of Section 7 and Appendix F of the article.

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.quadraticFeature_invariant.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (σ : ℝ → ℝ) (g : E ≃ᵃ[ℝ] E) (x : E)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticFeature σ (g x)
      ((LeanRidgelet.quadraticParameterLinearEquiv g) ξ) =
    LeanRidgelet.quadraticFeature σ x ξ

theorem LeanRidgelet.quadraticFeature_invariant.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (σ : ℝ → ℝ)
  (g : E ≃ᵃ[ℝ] E) (x : E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticFeature σ (g x)
      ((LeanRidgelet.quadraticParameterLinearEquiv
          g)
        ξ) =
    LeanRidgelet.quadraticFeature σ x ξ

The quadratic feature is invariant under the joint affine data/parameter action.

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.quadraticArgument_parameter_injective.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  Function.Injective fun ξ x ↦ LeanRidgelet.quadraticArgument x ξ

theorem LeanRidgelet.quadraticArgument_parameter_injective.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  Function.Injective fun ξ x ↦
    LeanRidgelet.quadraticArgument x ξ

Quadratic parameters are determined by their scalar functional: the constant is the value at
the origin, the odd part determines the linear coefficient, and the even part determines the
quadratic form, hence the symmetric coefficient by polarization.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticParameterSMul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  SMul (E ≃ᵃ[ℝ] E) (LeanRidgelet.QuadraticParameter E)

def LeanRidgelet.quadraticParameterSMul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  SMul (E ≃ᵃ[ℝ] E)
    (LeanRidgelet.QuadraticParameter E)

Implementation after :=

:= 2000) quadraticParameterSMul :
    SMul (E ≃ᵃ[ℝ] E) (QuadraticParameter E) :=
  ⟨fun g ξ ↦ quadraticParameterLinearEquiv g ξ⟩

The joint affine action on quadratic parameters.  The priority keeps the componentwise
product action, which does not exist here, from being searched first.

defdefined in LeanRidgelet/HA/Quadratic.lean

complete

def LeanRidgelet.quadraticParameterMulAction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  MulAction (E ≃ᵃ[ℝ] E) (LeanRidgelet.QuadraticParameter E)

def LeanRidgelet.quadraticParameterMulAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  MulAction (E ≃ᵃ[ℝ] E)
    (LeanRidgelet.QuadraticParameter E)

Implementation after :=

:= by
    apply quadraticArgument_parameter_injective
    funext x
    change quadraticArgument x (quadraticParameterLinearEquiv (AffineEquiv.refl ℝ E) ξ) =
      quadraticArgument x ξ
    simpa using quadraticArgument_invariant (AffineEquiv.refl ℝ E) x ξ
  mul_smul g h ξ := by
    apply quadraticArgument_parameter_injective
    funext x
    have hgh := quadraticArgument_invariant (g * h) ((g * h).symm x) ξ
    have hg := quadraticArgument_invariant g (g.symm x) (quadraticParameterLinearEquiv h ξ)
    have hh := quadraticArgument_invariant h (h.symm (g.symm x)) ξ
    have hgh' : quadraticArgument x (quadraticParameterLinearEquiv (g * h) ξ) =
        quadraticArgument ((g * h).symm x) ξ := by simpa using hgh
    have hg' : quadraticArgument x
          (quadraticParameterLinearEquiv g (quadraticParameterLinearEquiv h ξ)) =
        quadraticArgument (g.symm x) (quadraticParameterLinearEquiv h ξ) := by
      simpa using hg
    have hh' : quadraticArgument (g.symm x) (quadraticParameterLinearEquiv h ξ) =
        quadraticArgument (h.symm (g.symm x)) ξ := by simpa using hh
    exact hgh'.trans <| by
      rw [show (g * h).symm x = h.symm (g.symm x) by rfl]
      exact (hg'.trans hh').symm

The joint affine action on quadratic parameters.

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.quadraticParameter_smul_def.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (ξ : LeanRidgelet.QuadraticParameter E) :
  g • ξ = (LeanRidgelet.quadraticParameterLinearEquiv g) ξ

theorem LeanRidgelet.quadraticParameter_smul_def.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  g • ξ =
    (LeanRidgelet.quadraticParameterLinearEquiv
        g)
      ξ

theoremdefined in LeanRidgelet/HA/Quadratic.lean

complete

theorem LeanRidgelet.det_quadraticParameterLinearEquiv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LinearMap.det ↑(LeanRidgelet.quadraticParameterLinearEquiv g) =
    LinearMap.det ↑(LeanRidgelet.quadraticCongr g.linear.symm) *
      (LinearMap.det ↑g.linear)⁻¹

theorem LeanRidgelet.det_quadraticParameterLinearEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LinearMap.det
      ↑(LeanRidgelet.quadraticParameterLinearEquiv
          g) =
    LinearMap.det
        ↑(LeanRidgelet.quadraticCongr
            g.linear.symm) *
      (LinearMap.det ↑g.linear)⁻¹

The determinant of the quadratic parameter action is the determinant of its symmetric
congruence block times the reciprocal determinant of the linear part.

Symmetric matrices are self-adjoint continuous endomorphisms of the data space, so no basis is chosen and the transpose is the adjoint; they form a real subspace, which is what later puts a parameter measure on them. Writing s=L^{-1}t, congruence turns every term into an inner product at the shifted point: the quadratic term becomes \langle x+s, A(x+s)\rangle, the linear term \langle x+s, b-2As\rangle, and the constant c-\langle s,b\rangle+\langle s,As\rangle. Expanding, the cross terms cancel — this is where symmetry of A is used, through \langle s,Ax\rangle=\langle x,As\rangle — and the scalar argument of the activation is left invariant. Hence so is the feature, for every activation.

The parameter action is linear, not merely affine, and it is block lower triangular for the filtration by the symmetric coefficient and then the linear one: its determinant is the determinant of the congruence A\mapsto L^{-\top}AL^{-1} on symmetric coefficients times (\det L)^{-1}. The congruence determinant is left abstract, since the parameter measure only needs it to be a nonzero constant; its explicit value (\det L)^{-(m+1)} is not used. That the action is a group action is not proved by composing the three block formulas but obtained from the invariance itself: a quadratic parameter is determined by its scalar functional — the constant is the value at the origin, the odd part determines the linear coefficient, and the even part determines the quadratic form, hence the symmetric coefficient by polarization — so two parameter transformations agreeing on all functionals are equal.

Theorem4.5.2

Statement uses 2

Statement dependency previews

Preview

Lemma 4.5.1

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

used by 1✓L∃∀N

Appendix C at Section 7. The synthesis integral over a compact family of quadratic parameters is approximated, to any accuracy and uniformly in the data, by a finite network with nonnegative weights.

Lean code for Theorem4.5.2●4 declarations

theoremdefined in LeanRidgelet/HA/QuadraticFiniteWidth.lean

complete

theorem LeanRidgelet.continuous_quadraticVectorFeature.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {σ : ℝ → ℂ} (hσ : Continuous σ)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  Continuous fun x ↦ LeanRidgelet.quadraticVectorFeature σ x ξ

theorem LeanRidgelet.continuous_quadraticVectorFeature.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {σ : ℝ → ℂ}
  (hσ : Continuous σ)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  Continuous fun x ↦
    LeanRidgelet.quadraticVectorFeature σ
      x ξ

The quadratic feature at a fixed parameter is continuous in the data: the scalar argument is
jointly continuous, and the activation is continuous.

defdefined in LeanRidgelet/HA/QuadraticFiniteWidth.lean

complete

def LeanRidgelet.quadraticFeatureBoundedContinuous.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {σ : ℝ → ℂ} (hσc : Continuous σ) {C : ℝ} (hσb : ∀ (z : ℝ), ‖σ z‖ ≤ C)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  BoundedContinuousFunction E ℂ

def LeanRidgelet.quadraticFeatureBoundedContinuous.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {σ : ℝ → ℂ}
  (hσc : Continuous σ) {C : ℝ}
  (hσb : ∀ (z : ℝ), ‖σ z‖ ≤ C)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  BoundedContinuousFunction E ℂ

Implementation after :=

:=
  BoundedContinuousFunction.mkOfBound
    ⟨fun x ↦ quadraticVectorFeature σ x ξ, continuous_quadraticVectorFeature hσc ξ⟩ (2 * C)
    fun x y ↦ by
      calc dist (quadraticVectorFeature σ x ξ) (quadraticVectorFeature σ y ξ)
          = ‖quadraticVectorFeature σ x ξ - quadraticVectorFeature σ y ξ‖ := dist_eq_norm _ _
        _ ≤ ‖quadraticVectorFeature σ x ξ‖ + ‖quadraticVectorFeature σ y ξ‖ := norm_sub_le _ _
        _ ≤ C + C := add_le_add (hσb _) (hσb _)
        _ = 2 * C := by ring

**The quadratic feature as a bounded continuous function of the data.**  For a bounded
continuous
activation the feature at a fixed parameter is an element of the space whose norm is the supremum
norm, which is the space Appendix C's approximation is uniform in.

theoremdefined in LeanRidgelet/HA/QuadraticFiniteWidth.lean

complete

theorem LeanRidgelet.quadraticFeatureBoundedContinuous_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {σ : ℝ → ℂ} (hσc : Continuous σ) {C : ℝ}
  (hσb : ∀ (z : ℝ), ‖σ z‖ ≤ C) (ξ : LeanRidgelet.QuadraticParameter E)
  (x : E) :
  (LeanRidgelet.quadraticFeatureBoundedContinuous hσc hσb ξ) x =
    LeanRidgelet.quadraticVectorFeature σ x ξ

theorem LeanRidgelet.quadraticFeatureBoundedContinuous_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {σ : ℝ → ℂ}
  (hσc : Continuous σ) {C : ℝ}
  (hσb : ∀ (z : ℝ), ‖σ z‖ ≤ C)
  (ξ : LeanRidgelet.QuadraticParameter E)
  (x : E) :
  (LeanRidgelet.quadraticFeatureBoundedContinuous
        hσc hσb ξ)
      x =
    LeanRidgelet.quadraticVectorFeature σ
      x ξ

theoremdefined in LeanRidgelet/HA/QuadraticFiniteWidth.lean

complete

theorem LeanRidgelet.exists_finite_quadraticNetwork_approx.{u_1, u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {σ : ℝ → ℂ} (hσc : Continuous σ) {C : ℝ}
  (hσb : ∀ (z : ℝ), ‖σ z‖ ≤ C) {Ξ : Type u_2} [MetricSpace Ξ]
  [CompactSpace Ξ] [MeasurableSpace Ξ] [BorelSpace Ξ]
  (κ : MeasureTheory.Measure Ξ) [MeasureTheory.IsFiniteMeasure κ]
  (ι : Ξ → LeanRidgelet.QuadraticParameter E) (γ : Ξ → ℂ) {L : NNReal}
  (hlip :
    LipschitzWith L fun t ↦
      γ t •
        LeanRidgelet.quadraticFeatureBoundedContinuous hσc hσb (ι t))
  {ε : ℝ} (hε : 0 < ε) :
  ∃ n w t,
    (∀ (i : Fin n), 0 ≤ w i) ∧
      ∀ (x : E),
        ‖∫ (s : Ξ),
                γ s • LeanRidgelet.quadraticVectorFeature σ x (ι s) ∂κ -
              ∑ i,
                w i •
                  γ (t i) •
                    LeanRidgelet.quadraticVectorFeature σ x (ι (t i))‖ <
          ε

theorem LeanRidgelet.exists_finite_quadraticNetwork_approx.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {σ : ℝ → ℂ}
  (hσc : Continuous σ) {C : ℝ}
  (hσb : ∀ (z : ℝ), ‖σ z‖ ≤ C)
  {Ξ : Type u_2} [MetricSpace Ξ]
  [CompactSpace Ξ] [MeasurableSpace Ξ]
  [BorelSpace Ξ]
  (κ : MeasureTheory.Measure Ξ)
  [MeasureTheory.IsFiniteMeasure κ]
  (ι :
    Ξ → LeanRidgelet.QuadraticParameter E)
  (γ : Ξ → ℂ) {L : NNReal}
  (hlip :
    LipschitzWith L fun t ↦
      γ t •
        LeanRidgelet.quadraticFeatureBoundedContinuous
          hσc hσb (ι t))
  {ε : ℝ} (hε : 0 < ε) :
  ∃ n w t,
    (∀ (i : Fin n), 0 ≤ w i) ∧
      ∀ (x : E),
        ‖∫ (s : Ξ),
                γ s •
                  LeanRidgelet.quadraticVectorFeature
                    σ x (ι s) ∂κ -
              ∑ i,
                w i •
                  γ (t i) •
                    LeanRidgelet.quadraticVectorFeature
                      σ x (ι (t i))‖ <
          ε

**Uniform approximation of the quadratic-form network by a finite network.**  The synthesis
integral of a coefficient function over a compact family of quadratic parameters is approximated, to
any accuracy and uniformly in the data, by a finite sum of features at finitely many parameters with
nonnegative weights.  This is Appendix C of the article at the Section 7 architecture.

The hypotheses are those of the general theorem: the activation is continuous and bounded, the index
space is compact metric with a finite Borel measure, and the coefficient-weighted family is
Lipschitz
in the index.  The conclusion is stated pointwise in the data, which is what the supremum norm of
the
approximating space gives.

The general theorem is in the Mathlib candidate layer: a Lipschitz family of bounded continuous functions indexed by a compact metric space with a finite measure has its Bochner integral as a uniform limit of nonnegative-weighted finite sums. Instantiating it here needs two conditions on the activation, and both are the usual ones for a uniform-approximation theorem. Continuity, so that the feature is continuous in the data; and boundedness, so that the feature is an element of the space whose norm is the supremum norm.

An activation of polynomial growth is excluded, and for a reason worth stating plainly rather than working around: x\mapsto\sigma(q_\xi(x)) is then unbounded, so it is not an element of that space at all, and a uniform-in-x approximation over an unbounded data space is not the statement one would want for it. The rectified linear unit belongs to the boundedness discussion below, not to this one.

The parameter side is indexed by an abstract compact metric space mapping into the parameter space, because the relatively invariant parameter measure is neither compact nor finite. That is the honest form of the statement: a finite network is a finite sum of features at finitely many parameters, and the theorem says the integral over any compact family is approximated by such a sum.

Theorem4.5.3

Statement uses 3

Statement dependency previews

Preview

Definition 4.2.5

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

used by 1✓L∃∀N

Quadratic parameter measure. The quasi-invariant L² representation of the affine group on quadratic parameters, with the constant Radon--Nikodym density of the parameter action.

Lean code for Theorem4.5.3●13 declarations

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameterLinearEquiv_one.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  LeanRidgelet.quadraticParameterLinearEquiv 1 =
    LinearEquiv.refl ℝ (LeanRidgelet.QuadraticParameter E)

theorem LeanRidgelet.quadraticParameterLinearEquiv_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  LeanRidgelet.quadraticParameterLinearEquiv
      1 =
    LinearEquiv.refl ℝ
      (LeanRidgelet.QuadraticParameter E)

The identity of the affine group acts on quadratic parameters by the identity.  This is the
group action law `one_smul` read back through `LeanRidgelet.quadraticParameter_smul_def`.

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameterLinearEquiv_mul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g h : E ≃ᵃ[ℝ] E) :
  ↑(LeanRidgelet.quadraticParameterLinearEquiv (g * h)) =
    ↑(LeanRidgelet.quadraticParameterLinearEquiv g) ∘ₗ
      ↑(LeanRidgelet.quadraticParameterLinearEquiv h)

theorem LeanRidgelet.quadraticParameterLinearEquiv_mul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g h : E ≃ᵃ[ℝ] E) :
  ↑(LeanRidgelet.quadraticParameterLinearEquiv
        (g * h)) =
    ↑(LeanRidgelet.quadraticParameterLinearEquiv
          g) ∘ₗ
      ↑(LeanRidgelet.quadraticParameterLinearEquiv
          h)

The parameter action is multiplicative.  This is the group action law `mul_smul` read back
through `LeanRidgelet.quadraticParameter_smul_def`; no block computation is needed.

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameterLinearEquiv_inv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.quadraticParameterLinearEquiv g⁻¹ =
    (LeanRidgelet.quadraticParameterLinearEquiv g).symm

theorem LeanRidgelet.quadraticParameterLinearEquiv_inv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.quadraticParameterLinearEquiv
      g⁻¹ =
    (LeanRidgelet.quadraticParameterLinearEquiv
        g).symm

The inverse affine group element acts by the inverse parameter equivalence.

defdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

def LeanRidgelet.quadraticParameterJacobian.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : LeanRidgelet.QuadraticParameter E → NNReal

def LeanRidgelet.quadraticParameterJacobian.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.QuadraticParameter E →
    NNReal

Implementation after :=

:=
  fun _ ↦ ‖LinearMap.det
    (quadraticParameterLinearEquiv g : QuadraticParameter E →ₗ[ℝ] QuadraticParameter E)‖₊

The constant Radon--Nikodym density for the inverse quadratic parameter action.  It is the
absolute determinant of the parameter equivalence, left unevaluated.

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameterJacobian_one.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticParameterJacobian 1 ξ = 1

theorem LeanRidgelet.quadraticParameterJacobian_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticParameterJacobian
      1 ξ =
    1

The parameter Jacobian is normalized at the identity.

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameterJacobian_cocycle.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g h : E ≃ᵃ[ℝ] E) (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticParameterJacobian (g * h) ξ =
    LeanRidgelet.quadraticParameterJacobian g (h • ξ) *
      LeanRidgelet.quadraticParameterJacobian h ξ

theorem LeanRidgelet.quadraticParameterJacobian_cocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g h : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticParameterJacobian
      (g * h) ξ =
    LeanRidgelet.quadraticParameterJacobian
        g (h • ξ) *
      LeanRidgelet.quadraticParameterJacobian
        h ξ

The parameter Jacobian obeys the Radon--Nikodym cocycle law.

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameterJacobian_ne_zero.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticParameterJacobian g ξ ≠ 0

theorem LeanRidgelet.quadraticParameterJacobian_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticParameterJacobian
      g ξ ≠
    0

The parameter Jacobian is everywhere nonzero, because the parameter action is by linear
equivalences and the determinant of a linear equivalence is a unit.

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameterJacobian_eq_blocks.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticParameterJacobian g ξ =
    ‖LinearMap.det ↑(LeanRidgelet.quadraticCongr g.linear.symm)‖₊ *
      ‖LinearMap.det ↑g.linear‖₊⁻¹

theorem LeanRidgelet.quadraticParameterJacobian_eq_blocks.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticParameterJacobian
      g ξ =
    ‖LinearMap.det
          ↑(LeanRidgelet.quadraticCongr
              g.linear.symm)‖₊ *
      ‖LinearMap.det ↑g.linear‖₊⁻¹

The block form of the parameter Jacobian: the absolute determinant of the symmetric congruence
block times the reciprocal absolute determinant of the linear part.  The second factor is the ridge
parameter Jacobian, so the quadratic density differs from the ridge one exactly by the congruence
block.  Nothing below uses this identity; it records the article's factorization.

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameter_measurable.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] (g : E ≃ᵃ[ℝ] E) :
  Measurable fun ξ ↦ g • ξ

theorem LeanRidgelet.quadraticParameter_measurable.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (g : E ≃ᵃ[ℝ] E) :
  Measurable fun ξ ↦ g • ξ

The quadratic parameter action is measurable, being linear on a finite-dimensional space.

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameterJacobian_measurable.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (g : E ≃ᵃ[ℝ] E) :
  Measurable (LeanRidgelet.quadraticParameterJacobian g)

theorem LeanRidgelet.quadraticParameterJacobian_measurable.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (g : E ≃ᵃ[ℝ] E) :
  Measurable
    (LeanRidgelet.quadraticParameterJacobian
      g)

The constant parameter Jacobian is measurable.

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameter_map_eq_withDensity.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (ν : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [ν.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) :
  MeasureTheory.Measure.map
      (⇑(LeanRidgelet.quadraticParameterLinearEquiv g).symm) ν =
    ν.withDensity fun ξ ↦ ↑(LeanRidgelet.quadraticParameterJacobian g ξ)

theorem LeanRidgelet.quadraticParameter_map_eq_withDensity.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (ν :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [ν.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) :
  MeasureTheory.Measure.map
      (⇑(LeanRidgelet.quadraticParameterLinearEquiv
            g).symm)
      ν =
    ν.withDensity fun ξ ↦
      ↑(LeanRidgelet.quadraticParameterJacobian
          g ξ)

Pushforward of parameter Haar measure by the inverse parameter equivalence.

theoremdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

theorem LeanRidgelet.quadraticParameter_group_map_eq_withDensity.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (ν : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [ν.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) :
  MeasureTheory.Measure.map (fun ξ ↦ g⁻¹ • ξ) ν =
    ν.withDensity fun ξ ↦ ↑(LeanRidgelet.quadraticParameterJacobian g ξ)

theorem LeanRidgelet.quadraticParameter_group_map_eq_withDensity.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (ν :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [ν.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) :
  MeasureTheory.Measure.map
      (fun ξ ↦ g⁻¹ • ξ) ν =
    ν.withDensity fun ξ ↦
      ↑(LeanRidgelet.quadraticParameterJacobian
          g ξ)

The parameter pushforward formula expressed through the affine-group action instance.  This is
the strong quasi-invariance hypothesis of the generic `L²` construction.

defdefined in LeanRidgelet/HA/QuadraticMeasure.lean

complete

def LeanRidgelet.quadraticParameterLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (ν : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [ν.IsAddHaarMeasure] [MeasureTheory.SigmaFinite ν] :
  UnitaryRepresentation (E ≃ᵃ[ℝ] E) ↥(MeasureTheory.Lp ℂ 2 ν)

def LeanRidgelet.quadraticParameterLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (ν :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [ν.IsAddHaarMeasure]
  [MeasureTheory.SigmaFinite ν] :
  UnitaryRepresentation (E ≃ᵃ[ℝ] E)
    ↥(MeasureTheory.Lp ℂ 2 ν)

Implementation after :=

:=
  quasiInvariantLpUnitaryRepresentation quadraticParameterJacobian
    quadraticParameter_measurable (quadraticParameter_group_map_eq_withDensity ν)
    quadraticParameterJacobian_measurable quadraticParameterJacobian_ne_zero
    quadraticParameterJacobian_one quadraticParameterJacobian_cocycle

The Radon--Nikodym-corrected quadratic parameter representation on scalar `L²`.

The parameter action is a linear automorphism of a finite-dimensional space, so pushing an additive Haar measure forward along it multiplies it by the absolute determinant, a nonzero constant. That determinant is exactly the density, so the Jacobian is defined intrinsically as the norm of the determinant of the parameter equivalence and never has to be evaluated; the ridge case is the same quantity written through the data determinant, since there the parameter determinant is (\det L)^{-1}. The cocycle law comes from multiplicativity of the determinant along the composition law of the action, and that composition law is read off the group-action instance rather than recomputed from the three blocks. Non-vanishing is that the determinant of an equivalence is a unit, so the density is admissible for the quasi-invariant construction, which then supplies the unitary representation. The parameter space carries no global measurable structure: the measurable space and Borel structure of the symmetric coefficients are assumed, exactly as the data space assumes its own, and the product structure follows.

Here the quadratic feature parts company with the ridge one. The generic quasi-invariant synthesis and ridgelet identities require the parameter density to be the reciprocal of the data density, and for ridge parameters that holds exactly, the parameter determinant being (\det L)^{-1} against the data determinant \det L. For quadratic parameters the congruence factor survives, so the two densities differ by a positive character of the affine group, and a synthesis identity intertwines the parameter representation with a character-twisted data action rather than with the data representation. The twist does not cancel in the composite of synthesis with ridgelet: both halves pick up the same factor rather than reciprocal ones, so the composite carries its square, and a square equal to one forces the character to be one — that is, exact balance. So the twist has to be removed rather than tolerated, and the next node does that by changing the parameter measure. The article does not meet this point because it states the example in invariant-measure notation.

Theorem4.5.4

Statement uses 3

Statement dependency previews

Preview

Theorem 4.5.3

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

Used by 3

Reverse dependency previews

Preview

Theorem 4.5.5

Loading preview

Reverse dependency preview content is loaded from the rendered-fragment cache.

✓L∃∀N

The balanced parameter measure of Section 7. Weighting the parameter measure by a power of the determinant of the symmetric coefficient restores the balance the quadratic feature otherwise fails, and the two quasi-invariant Bochner identities then hold for it in their untwisted form.

Lean code for Theorem4.5.4●24 declarations

defdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

def LeanRidgelet.quadraticSymmetricEquivSelfAdjoint.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  LeanRidgelet.QuadraticSymmetric E ≃ₗ[ℝ] ↥(selfAdjoint (E →L[ℝ] E))

def LeanRidgelet.quadraticSymmetricEquivSelfAdjoint.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  LeanRidgelet.QuadraticSymmetric E ≃ₗ[ℝ]
    ↥(selfAdjoint (E →L[ℝ] E))

Implementation after :=

:= ⟨(A : E →L[ℝ] E), A.2⟩
  map_add' _ _ := rfl
  map_smul' _ _ := rfl
  invFun A := ⟨(A : E →L[ℝ] E), A.2⟩
  left_inv _ := rfl
  right_inv _ := rfl

The symmetric coefficients of a quadratic parameter are the self-adjoint continuous
endomorphisms.  The two are the same subtype with the same module structure, but the instances are
found by different paths, so the identity is bundled here once.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.det_quadraticCongr.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (M : E ≃ₗ[ℝ] E) :
  LinearMap.det ↑(LeanRidgelet.quadraticCongr M) =
    LinearMap.det ↑M ^ (Module.finrank ℝ E + 1)

theorem LeanRidgelet.det_quadraticCongr.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (M : E ≃ₗ[ℝ] E) :
  LinearMap.det
      ↑(LeanRidgelet.quadraticCongr M) =
    LinearMap.det ↑M ^
      (Module.finrank ℝ E + 1)

The determinant of the congruence on symmetric coefficients is the power `m + 1` of the
determinant of the conjugating map.  This is the congruence-determinant input of the relative
measure, discharged by the general matrix computation of
`Matrix.det_congrMap`: the symmetric coefficients of a quadratic parameter are exactly the
self-adjoint continuous endomorphisms, with the same module structure, so the two congruence maps
agree on the nose.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.det_quadraticCongr_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (M : E ≃ₗ[ℝ] E) (A : LeanRidgelet.QuadraticSymmetric E) :
  (↑((LeanRidgelet.quadraticCongr M) A)).det =
    LinearMap.det ↑M ^ 2 * (↑A).det

theorem LeanRidgelet.det_quadraticCongr_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (M : E ≃ₗ[ℝ] E)
  (A :
    LeanRidgelet.QuadraticSymmetric E) :
  (↑((LeanRidgelet.quadraticCongr M)
          A)).det =
    LinearMap.det ↑M ^ 2 * (↑A).det

**The relative invariant.**  Congruence by `M` multiplies the determinant of a symmetric
coefficient by `(det M) ^ 2`, because the two outer factors are adjoint to each other and an
adjoint has the same determinant.

defdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

def LeanRidgelet.quadraticSymmetricDet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (ξ : LeanRidgelet.QuadraticParameter E) : ℝ

def LeanRidgelet.quadraticSymmetricDet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  ℝ

Implementation after :=

:= (ξ.1 : E →L[ℝ] E).det

The determinant of the symmetric coefficient of a quadratic parameter.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticSymmetricDet_smul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticSymmetricDet (g • ξ) =
    (LinearMap.det ↑g.linear)⁻¹ ^ 2 *
      LeanRidgelet.quadraticSymmetricDet ξ

theorem LeanRidgelet.quadraticSymmetricDet_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticSymmetricDet
      (g • ξ) =
    (LinearMap.det ↑g.linear)⁻¹ ^ 2 *
      LeanRidgelet.quadraticSymmetricDet ξ

**Relative invariance of the symmetric determinant.**  The parameter action of `g` acts on the
symmetric block by congruence with `g.linear.symm`, so it multiplies the symmetric determinant by
the square of the reciprocal determinant of the linear part.

defdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

def LeanRidgelet.quadraticNondegenerate.{u_2} (E : Type u_2)
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] : Set (LeanRidgelet.QuadraticParameter E)

def LeanRidgelet.quadraticNondegenerate.{u_2}
  (E : Type u_2) [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  Set (LeanRidgelet.QuadraticParameter E)

Implementation after :=

:=
  {ξ | quadraticSymmetricDet ξ ≠ 0}

The nondegenerate locus of quadratic parameters, that is, those whose symmetric coefficient is
invertible.  This is the support of the relatively invariant measure below.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.preimage_smul_quadraticNondegenerate.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  (fun ξ ↦ g • ξ) ⁻¹' LeanRidgelet.quadraticNondegenerate E =
    LeanRidgelet.quadraticNondegenerate E

theorem LeanRidgelet.preimage_smul_quadraticNondegenerate.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  (fun ξ ↦ g • ξ) ⁻¹'
      LeanRidgelet.quadraticNondegenerate
        E =
    LeanRidgelet.quadraticNondegenerate E

**Invariance of the degenerate locus.**  The nondegenerate locus is exactly its own preimage
under the parameter action, so restricting a measure to it is compatible with the action.

defdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

def LeanRidgelet.quadraticRelativeWeight.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (ξ : LeanRidgelet.QuadraticParameter E) : ENNReal

def LeanRidgelet.quadraticRelativeWeight.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  ENNReal

Implementation after :=

:=
  ENNReal.ofReal (|quadraticSymmetricDet ξ| ^ (-(Module.finrank ℝ E + 1 : ℝ) / 2))

The weight defining the relatively invariant parameter measure: the power
`-(m + 1) / 2` of the absolute symmetric determinant, where `m = Module.finrank ℝ E`.  The
exponent is chosen so that the weight transforms by `‖det L‖₊ ^ (m + 1)`, which is exactly what
cancels the congruence factor in the parameter Jacobian.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticRelativeWeight_smul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticRelativeWeight (g • ξ) =
    ↑‖LinearMap.det ↑g.linear‖₊ ^ (Module.finrank ℝ E + 1) *
      LeanRidgelet.quadraticRelativeWeight ξ

theorem LeanRidgelet.quadraticRelativeWeight_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticRelativeWeight
      (g • ξ) =
    ↑‖LinearMap.det ↑g.linear‖₊ ^
        (Module.finrank ℝ E + 1) *
      LeanRidgelet.quadraticRelativeWeight
        ξ

**The weight law.**  The parameter action of `g` multiplies the weight by
`‖det L‖₊ ^ (m + 1)`, where `L` is the linear part of `g`.  No nondegeneracy is needed: on the
degenerate locus both sides vanish.

defdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

def LeanRidgelet.quadraticRelativeParameterJacobian.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : LeanRidgelet.QuadraticParameter E → NNReal

def LeanRidgelet.quadraticRelativeParameterJacobian.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.QuadraticParameter E →
    NNReal

Implementation after :=

:=
  fun _ ↦ ‖LinearMap.det (g.linear : E →ₗ[ℝ] E)‖₊⁻¹

The constant Radon--Nikodym density of the inverse quadratic parameter action for the
relatively invariant measure.  Unlike `LeanRidgelet.quadraticParameterJacobian`, it is the exact
reciprocal of the data density `LeanRidgelet.affineDataJacobian`; it is literally the ridge
parameter Jacobian `LeanRidgelet.affineParameterJacobian` read on quadratic parameters.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticRelativeParameterJacobian_one.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticRelativeParameterJacobian 1 ξ = 1

theorem LeanRidgelet.quadraticRelativeParameterJacobian_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticRelativeParameterJacobian
      1 ξ =
    1

The relative parameter Jacobian is normalized at the identity.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticRelativeParameterJacobian_cocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g h : E ≃ᵃ[ℝ] E)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticRelativeParameterJacobian (g * h) ξ =
    LeanRidgelet.quadraticRelativeParameterJacobian g (h • ξ) *
      LeanRidgelet.quadraticRelativeParameterJacobian h ξ

theorem LeanRidgelet.quadraticRelativeParameterJacobian_cocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g h : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticRelativeParameterJacobian
      (g * h) ξ =
    LeanRidgelet.quadraticRelativeParameterJacobian
        g (h • ξ) *
      LeanRidgelet.quadraticRelativeParameterJacobian
        h ξ

The relative parameter Jacobian obeys the Radon--Nikodym cocycle law.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticRelativeParameterJacobian_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticRelativeParameterJacobian g ξ ≠ 0

theorem LeanRidgelet.quadraticRelativeParameterJacobian_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticRelativeParameterJacobian
      g ξ ≠
    0

The relative parameter Jacobian is everywhere nonzero.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticRelative_synthesis_radonNikodym_balance.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E) (x : E)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  ↑↑(NNReal.sqrt
          (LeanRidgelet.quadraticRelativeParameterJacobian g ξ)) =
    LeanRidgelet.radonNikodymWeight LeanRidgelet.affineDataJacobian g x

theorem LeanRidgelet.quadraticRelative_synthesis_radonNikodym_balance.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (x : E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  ↑↑(NNReal.sqrt
          (LeanRidgelet.quadraticRelativeParameterJacobian
            g ξ)) =
    LeanRidgelet.radonNikodymWeight
      LeanRidgelet.affineDataJacobian g x

The synthesis balance in the form expected by the generic quasi-invariant theorem.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticRelative_ridgelet_radonNikodym_balance.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E) (x : E)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  ↑↑(NNReal.sqrt (LeanRidgelet.affineDataJacobian g x)) =
    LeanRidgelet.radonNikodymWeight
      LeanRidgelet.quadraticRelativeParameterJacobian g ξ

theorem LeanRidgelet.quadraticRelative_ridgelet_radonNikodym_balance.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (x : E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  ↑↑(NNReal.sqrt
          (LeanRidgelet.affineDataJacobian
            g x)) =
    LeanRidgelet.radonNikodymWeight
      LeanRidgelet.quadraticRelativeParameterJacobian
      g ξ

The ridgelet balance in the form expected by the generic quasi-invariant theorem.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.nnnorm_det_quadraticParameterLinearEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E) :
  ‖LinearMap.det ↑(LeanRidgelet.quadraticParameterLinearEquiv g)‖₊ =
    (‖LinearMap.det ↑g.linear‖₊ ^ (Module.finrank ℝ E + 2))⁻¹

theorem LeanRidgelet.nnnorm_det_quadraticParameterLinearEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  ‖LinearMap.det
        ↑(LeanRidgelet.quadraticParameterLinearEquiv
            g)‖₊ =
    (‖LinearMap.det ↑g.linear‖₊ ^
        (Module.finrank ℝ E + 2))⁻¹

The absolute determinant of the quadratic parameter action: it is the reciprocal of
`‖det L‖₊ ^ (m + 2)`, where `L` is the linear part.

defdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

def LeanRidgelet.quadraticRelativeMeasure.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)) :
  MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)

def LeanRidgelet.quadraticRelativeMeasure.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter
        E)) :
  MeasureTheory.Measure
    (LeanRidgelet.QuadraticParameter E)

Implementation after :=

:=
  (lam.restrict (quadraticNondegenerate E)).withDensity quadraticRelativeWeight

**The relatively invariant parameter measure.**  An additive Haar measure `lam` on the
quadratic parameter space, restricted to the nondegenerate locus and weighted by
`LeanRidgelet.quadraticRelativeWeight`.  It is σ-finite but, for `m ≥ 1`, not locally finite; see
the module docstring.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticRelative_map_restrict.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) :
  MeasureTheory.Measure.map (fun ξ ↦ g⁻¹ • ξ)
      (lam.restrict (LeanRidgelet.quadraticNondegenerate E)) =
    ENNReal.ofReal
        |(LinearMap.det
              ↑(LeanRidgelet.quadraticParameterLinearEquiv g⁻¹))⁻¹| •
      lam.restrict (LeanRidgelet.quadraticNondegenerate E)

theorem LeanRidgelet.quadraticRelative_map_restrict.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) :
  MeasureTheory.Measure.map
      (fun ξ ↦ g⁻¹ • ξ)
      (lam.restrict
        (LeanRidgelet.quadraticNondegenerate
          E)) =
    ENNReal.ofReal
        |(LinearMap.det
              ↑(LeanRidgelet.quadraticParameterLinearEquiv
                  g⁻¹))⁻¹| •
      lam.restrict
        (LeanRidgelet.quadraticNondegenerate
          E)

The restricted Haar measure is rescaled by the inverse parameter action, the rescaling being
the reciprocal absolute determinant of that action.  This is where the invariance of the
nondegenerate locus is used.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticRelativeParameter_group_map_eq_withDensity.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) :
  MeasureTheory.Measure.map (fun ξ ↦ g⁻¹ • ξ)
      (LeanRidgelet.quadraticRelativeMeasure lam) =
    (LeanRidgelet.quadraticRelativeMeasure lam).withDensity fun ξ ↦
      ↑(LeanRidgelet.quadraticRelativeParameterJacobian g ξ)

theorem LeanRidgelet.quadraticRelativeParameter_group_map_eq_withDensity.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) :
  MeasureTheory.Measure.map
      (fun ξ ↦ g⁻¹ • ξ)
      (LeanRidgelet.quadraticRelativeMeasure
        lam) =
    (LeanRidgelet.quadraticRelativeMeasure
          lam).withDensity
      fun ξ ↦
      ↑(LeanRidgelet.quadraticRelativeParameterJacobian
          g ξ)

**The balance.**  The relatively invariant parameter measure is strongly quasi-invariant with
the constant Radon--Nikodym density `‖det L‖₊⁻¹`, which is the exact reciprocal of the data density
`LeanRidgelet.affineDataJacobian`.  The congruence factor of
`LeanRidgelet.quadraticParameterJacobian_eq_blocks` is cancelled by the weight, whose
transformation law contributes `‖det L‖₊ ^ (m + 1)` against the `‖det L‖₊ ^ (m + 2)` of the Haar
Jacobian.  Compare `LeanRidgelet.affineParameter_group_map_eq_withDensity`.

defdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

def LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] :
  UnitaryRepresentation (E ≃ᵃ[ℝ] E)
    ↥(MeasureTheory.Lp ℂ 2 (LeanRidgelet.quadraticRelativeMeasure lam))

def LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] :
  UnitaryRepresentation (E ≃ᵃ[ℝ] E)
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure
          lam))

Implementation after :=

:=
  quasiInvariantLpUnitaryRepresentation quadraticRelativeParameterJacobian
    quadraticParameter_measurable
    (quadraticRelativeParameter_group_map_eq_withDensity lam)
    quadraticRelativeParameterJacobian_measurable quadraticRelativeParameterJacobian_ne_zero
    quadraticRelativeParameterJacobian_one quadraticRelativeParameterJacobian_cocycle

The Radon--Nikodym-corrected quadratic parameter representation on scalar `L²`, built from the
relatively invariant measure.  Compare `LeanRidgelet.quadraticParameterLpUnitaryRepresentation`,
which uses the Haar measure and therefore a different, unbalanced, cocycle.

defdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

def LeanRidgelet.quadraticVectorFeature.{u_1, u_2} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {Y : Type u_2} (σ : ℝ → Y) (x : E)
  (ξ : LeanRidgelet.QuadraticParameter E) : Y

def LeanRidgelet.quadraticVectorFeature.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2}
  (σ : ℝ → Y) (x : E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  Y

Implementation after :=

:=
  σ (quadraticArgument x ξ)

A vector-valued quadratic-form feature, the analogue of `LeanRidgelet.affineFeature` for the
quadratic argument.  The Bochner synthesis and ridgelet transforms take values in a complex inner
product space, so the scalar `LeanRidgelet.quadraticFeature` is not directly usable there.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticVectorFeature_jointInvariant.{u_1, u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2} [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y] [CompleteSpace Y] (σ : ℝ → Y) (g : E ≃ᵃ[ℝ] E)
  (x : E) (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticVectorFeature σ (g • x) (g • ξ) =
    ↑(1 g) (LeanRidgelet.quadraticVectorFeature σ x ξ)

theorem LeanRidgelet.quadraticVectorFeature_jointInvariant.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2}
  [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  [CompleteSpace Y] (σ : ℝ → Y)
  (g : E ≃ᵃ[ℝ] E) (x : E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticVectorFeature σ
      (g • x) (g • ξ) =
    ↑(1 g)
      (LeanRidgelet.quadraticVectorFeature
        σ x ξ)

A vector-valued quadratic feature is jointly invariant; equivalently, it is jointly
equivariant for the trivial output representation.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticRelativeBochnerSynthesis_intertwines.{u_1, u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] {Y : Type u_2}
  [NormedAddCommGroup Y] [InnerProductSpace ℂ Y] [CompleteSpace Y]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (σ : ℝ → Y) (g : E ≃ᵃ[ℝ] E)
  (γ : LeanRidgelet.QuadraticParameter E → ℂ) (x : E) :
  LeanRidgelet.bochnerSynthesis
      (LeanRidgelet.quadraticRelativeMeasure lam)
      (LeanRidgelet.quadraticVectorFeature σ)
      (LeanRidgelet.quasiRegularAction
        (LeanRidgelet.radonNikodymWeight
          LeanRidgelet.quadraticRelativeParameterJacobian)
        g γ)
      x =
    LeanRidgelet.quasiUnitaryPullbackAction
      LeanRidgelet.affineDataJacobian 1 g
      (LeanRidgelet.bochnerSynthesis
        (LeanRidgelet.quadraticRelativeMeasure lam)
        (LeanRidgelet.quadraticVectorFeature σ) γ)
      x

theorem LeanRidgelet.quadraticRelativeBochnerSynthesis_intertwines.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {Y : Type u_2} [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  [CompleteSpace Y]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (σ : ℝ → Y)
  (g : E ≃ᵃ[ℝ] E)
  (γ :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (x : E) :
  LeanRidgelet.bochnerSynthesis
      (LeanRidgelet.quadraticRelativeMeasure
        lam)
      (LeanRidgelet.quadraticVectorFeature
        σ)
      (LeanRidgelet.quasiRegularAction
        (LeanRidgelet.radonNikodymWeight
          LeanRidgelet.quadraticRelativeParameterJacobian)
        g γ)
      x =
    LeanRidgelet.quasiUnitaryPullbackAction
      LeanRidgelet.affineDataJacobian 1 g
      (LeanRidgelet.bochnerSynthesis
        (LeanRidgelet.quadraticRelativeMeasure
          lam)
        (LeanRidgelet.quadraticVectorFeature
          σ)
        γ)
      x

**The payoff, synthesis half.**  The untwisted quasi-invariant Bochner synthesis identity for
the quadratic feature, available because the relatively invariant parameter measure balances the
data measure exactly.  Compare `LeanRidgelet.affineBochnerSynthesis_intertwines`.

theoremdefined in LeanRidgelet/HA/QuadraticRelativeMeasure.lean

complete

theorem LeanRidgelet.quadraticRelativeBochnerRidgelet_intertwines.{u_1, u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  {Y : Type u_2} [NormedAddCommGroup Y] [InnerProductSpace ℂ Y]
  [CompleteSpace Y] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure]
  (ψ : ℝ → Y) (g : E ≃ᵃ[ℝ] E) (f : E → Y)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.bochnerRidgelet μ (LeanRidgelet.quadraticVectorFeature ψ)
      (LeanRidgelet.quasiUnitaryPullbackAction
        LeanRidgelet.affineDataJacobian 1 g f)
      ξ =
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g
      (LeanRidgelet.bochnerRidgelet μ
        (LeanRidgelet.quadraticVectorFeature ψ) f)
      ξ

theorem LeanRidgelet.quadraticRelativeBochnerRidgelet_intertwines.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {Y : Type u_2} [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  [CompleteSpace Y]
  (μ : MeasureTheory.Measure E)
  [μ.IsAddHaarMeasure] (ψ : ℝ → Y)
  (g : E ≃ᵃ[ℝ] E) (f : E → Y)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.bochnerRidgelet μ
      (LeanRidgelet.quadraticVectorFeature
        ψ)
      (LeanRidgelet.quasiUnitaryPullbackAction
        LeanRidgelet.affineDataJacobian 1
        g f)
      ξ =
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g
      (LeanRidgelet.bochnerRidgelet μ
        (LeanRidgelet.quadraticVectorFeature
          ψ)
        f)
      ξ

**The payoff, ridgelet half.**  The untwisted quasi-invariant Bochner ridgelet identity for the
quadratic feature.  Only the data measure enters, so this half needs neither the parameter measure
nor the congruence-determinant hypothesis; what it needs is the parameter Jacobian of the
relatively invariant measure, through the balance
`LeanRidgelet.quadraticRelative_ridgelet_radonNikodym_balance`.  Compare
`LeanRidgelet.affineBochnerRidgelet_intertwines`.

The determinant of the symmetric coefficient is a relative invariant: congruence multiplies it by the square of the determinant of the conjugating map, so under the parameter action it is multiplied by the inverse square of the determinant of the linear part. Raising it to the power -(m+1)/2 therefore produces a weight that the action multiplies by |\det L|^{m+1}, and against the determinant of the parameter action, |\det L|^{-(m+2)}, that leaves exactly |\det L|^{-1} — the reciprocal of the data density, which is the balance. The exponent is forced: it is the only one that cancels, and it works for every dimension and every symmetric coefficient, with no case distinction.

Two things make this legitimate. The degenerate locus, where the weight blows up, is invariant under the action, so the measure is taken on its complement and the weight is finite and positive on the whole support; no null-set argument about a determinant hypersurface is needed. And the resulting measure is only σ-finite, not locally finite — for positive dimension the weight is not locally integrable across the degenerate locus — but nothing in the quasi-invariant construction or in the Bochner identities uses local finiteness, only σ-finiteness. The feature is re-bundled with values in a complex inner-product space, since the generic identities need that target, and its joint invariance is the scalar one read through the trivial output representation.

Theorem4.5.5

Statement uses 4

Statement dependency previews

Preview

Theorem 4.1.6

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

Used by 2

Reverse dependency previews

Preview

Theorem 4.5.6

Loading preview

Reverse dependency preview content is loaded from the rendered-fragment cache.

✓L∃∀N

Section 7, the endpoint. The reconstruction operator of the quadratic-form network is a scalar multiple of the identity, and a nonzero scalar makes the normalized quadratic ridgelet transform a right inverse of the quadratic machine.

Lean code for Theorem4.5.5●20 declarations

theoremdefined in LeanRidgelet/HA/QuadraticReconstruction.lean

complete

theorem LeanRidgelet.quadraticReconstructionOperator_eq_smul_id.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M :
    LeanRidgelet.JointEquivariantMachine
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation)
  (R :
    LeanRidgelet.JointEquivariantRidgelet
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation) :
  ∃ c,
    LeanRidgelet.jointReconstructionOperator M R =
      c •
        ContinuousLinearMap.id ℂ
          ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)

theorem LeanRidgelet.quadraticReconstructionOperator_eq_smul_id.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M :
    LeanRidgelet.JointEquivariantMachine
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation)
  (R :
    LeanRidgelet.JointEquivariantRidgelet
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation) :
  ∃ c,
    LeanRidgelet.jointReconstructionOperator
        M R =
      c •
        ContinuousLinearMap.id ℂ
          ↥(MeasureTheory.Lp ℂ 2
              MeasureTheory.volume)

The reconstruction operator of the quadratic-form network is a scalar multiple of the identity.
This is Schur's lemma applied to the affine data representation, whose topological irreducibility is
proved in `LeanRidgelet.HA.AffineMackey`.

theoremdefined in LeanRidgelet/HA/QuadraticReconstruction.lean

complete

theorem LeanRidgelet.quadraticNormalizedRidgelet_rightInverse.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M :
    LeanRidgelet.JointEquivariantMachine
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation)
  (R :
    LeanRidgelet.JointEquivariantRidgelet
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation)
  {c : ℂ}
  (hrec :
    LeanRidgelet.jointReconstructionOperator M R =
      c •
        ContinuousLinearMap.id ℂ
          ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume))
  (hc : c ≠ 0) :
  Function.RightInverse ⇑(c⁻¹ • R.toContinuousLinearMap) ⇑M

theorem LeanRidgelet.quadraticNormalizedRidgelet_rightInverse.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M :
    LeanRidgelet.JointEquivariantMachine
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation)
  (R :
    LeanRidgelet.JointEquivariantRidgelet
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation)
  {c : ℂ}
  (hrec :
    LeanRidgelet.jointReconstructionOperator
        M R =
      c •
        ContinuousLinearMap.id ℂ
          ↥(MeasureTheory.Lp ℂ 2
              MeasureTheory.volume))
  (hc : c ≠ 0) :
  Function.RightInverse
    ⇑(c⁻¹ • R.toContinuousLinearMap) ⇑M

A nonzero reconstruction scalar makes the normalized quadratic ridgelet transform a right
inverse of the quadratic machine.

theoremdefined in LeanRidgelet/HA/QuadraticReconstruction.lean

complete

theorem LeanRidgelet.quadratic_reconstruction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M :
    LeanRidgelet.JointEquivariantMachine
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation)
  (R :
    LeanRidgelet.JointEquivariantRidgelet
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation) :
  ∃ c,
    LeanRidgelet.jointReconstructionOperator M R =
        c •
          ContinuousLinearMap.id ℂ
            ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) ∧
      (c ≠ 0 →
        Function.RightInverse ⇑(c⁻¹ • R.toContinuousLinearMap) ⇑M)

theorem LeanRidgelet.quadratic_reconstruction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M :
    LeanRidgelet.JointEquivariantMachine
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation)
  (R :
    LeanRidgelet.JointEquivariantRidgelet
      (LeanRidgelet.affineDataLpUnitaryRepresentation
          MeasureTheory.volume).toContRepresentation
      (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
          lam).toContRepresentation) :
  ∃ c,
    LeanRidgelet.jointReconstructionOperator
          M R =
        c •
          ContinuousLinearMap.id ℂ
            ↥(MeasureTheory.Lp ℂ 2
                MeasureTheory.volume) ∧
      (c ≠ 0 →
        Function.RightInverse
          ⇑(c⁻¹ • R.toContinuousLinearMap)
          ⇑M)

Section 7 of the article: the quadratic-form network reconstructs the data space.  The
reconstruction operator is a scalar, and a nonzero scalar turns the normalized quadratic ridgelet
transform into a right inverse of the quadratic machine.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.affineData_quasiUnitaryPullbackAction_congr_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) {f₁ f₂ : E → ℂ}
  (hf : f₁ =ᵐ[MeasureTheory.volume] f₂) :
  LeanRidgelet.quasiUnitaryPullbackAction
      LeanRidgelet.affineDataJacobian 1 g f₁ =ᵐ[MeasureTheory.volume]
    LeanRidgelet.quasiUnitaryPullbackAction
      LeanRidgelet.affineDataJacobian 1 g f₂

theorem LeanRidgelet.affineData_quasiUnitaryPullbackAction_congr_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) {f₁ f₂ : E → ℂ}
  (hf : f₁ =ᵐ[MeasureTheory.volume] f₂) :
  LeanRidgelet.quasiUnitaryPullbackAction
      LeanRidgelet.affineDataJacobian 1 g
      f₁ =ᵐ[MeasureTheory.volume]
    LeanRidgelet.quasiUnitaryPullbackAction
      LeanRidgelet.affineDataJacobian 1 g
      f₂

The corrected affine data action respects almost-everywhere equality of representatives, because
the inverse action is quasi measure preserving for the data measure.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.quadraticRelative_quasiRegularAction_congr_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  {γ₁ γ₂ : LeanRidgelet.QuadraticParameter E → ℂ}
  (hγ : γ₁ =ᵐ[LeanRidgelet.quadraticRelativeMeasure lam] γ₂) :
  LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g γ₁ =ᵐ[LeanRidgelet.quadraticRelativeMeasure lam]
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g γ₂

theorem LeanRidgelet.quadraticRelative_quasiRegularAction_congr_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  {γ₁ γ₂ :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hγ :
    γ₁ =ᵐ[LeanRidgelet.quadraticRelativeMeasure
        lam]
      γ₂) :
  LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g
      γ₁ =ᵐ[LeanRidgelet.quadraticRelativeMeasure
      lam]
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g γ₂

The quadratic parameter action respects almost-everywhere equality of representatives.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.quasiUnitaryPullbackAction_one_eq.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  (jac : (E ≃ᵃ[ℝ] E) → E → NNReal) (g : E ≃ᵃ[ℝ] E) (f : E → ℂ) :
  LeanRidgelet.quasiUnitaryPullbackAction jac 1 g f =
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight jac) g f

theorem LeanRidgelet.quasiUnitaryPullbackAction_one_eq.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  (jac : (E ≃ᵃ[ℝ] E) → E → NNReal)
  (g : E ≃ᵃ[ℝ] E) (f : E → ℂ) :
  LeanRidgelet.quasiUnitaryPullbackAction
      jac 1 g f =
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        jac)
      g f

With the trivial output representation the corrected pullback action is the corrected regular
action.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  (γ :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure lam))) :
  ↑↑(↑((LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
                lam)
              g)
          γ) =ᵐ[LeanRidgelet.quadraticRelativeMeasure lam]
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g fun ξ ↦ ↑↑γ ξ

theorem LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  (γ :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure
          lam))) :
  ↑↑(↑((LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
                lam)
              g)
          γ) =ᵐ[LeanRidgelet.quadraticRelativeMeasure
      lam]
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g fun ξ ↦ ↑↑γ ξ

The almost-everywhere formula for the quadratic parameter representation.

defdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

def LeanRidgelet.quadraticSynthesisKernel.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (σ : ℝ → ℂ) : E × LeanRidgelet.QuadraticParameter E → ℂ

def LeanRidgelet.quadraticSynthesisKernel.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (σ : ℝ → ℂ) :
  E × LeanRidgelet.QuadraticParameter E →
    ℂ

Implementation after :=

:=
  fun p ↦ conj (quadraticVectorFeature σ p.1 p.2)

The synthesis kernel of the quadratic feature: the conjugate feature, so that the kernel
operator integrates the coefficient against the feature itself.

defdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

def LeanRidgelet.quadraticAnalysisKernel.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (ψ : ℝ → ℂ) : LeanRidgelet.QuadraticParameter E × E → ℂ

def LeanRidgelet.quadraticAnalysisKernel.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (ψ : ℝ → ℂ) :
  LeanRidgelet.QuadraticParameter E × E →
    ℂ

Implementation after :=

:=
  fun p ↦ quadraticVectorFeature ψ p.2 p.1

The analysis kernel of the quadratic feature: the feature with its arguments exchanged, so that
the kernel operator pairs the data against the feature.

defdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

def LeanRidgelet.quadraticMachine.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticSynthesisKernel σ) 2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure lam))) :
  ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure lam)) →L[ℂ]
    ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)

def LeanRidgelet.quadraticMachine.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticSynthesisKernel
        σ)
      2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure
          lam))) :
  ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure
          lam)) →L[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)

Implementation after :=

:= E) σ) 2
      ((volume : Measure E).prod (quadraticRelativeMeasure lam))) :
    Lp ℂ 2 (quadraticRelativeMeasure lam) →L[ℂ] Lp ℂ 2 (volume : Measure E) :=
  hilbertSchmidtKernelOperator hσ

The quadratic machine: the synthesis integral of the quadratic feature, as a bounded operator
from parameter `L²` to data `L²`.

defdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

def LeanRidgelet.quadraticRidgelet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (hψ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticAnalysisKernel ψ) 2
      ((LeanRidgelet.quadraticRelativeMeasure lam).prod
        MeasureTheory.volume)) :
  ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) →L[ℂ]
    ↥(MeasureTheory.Lp ℂ 2 (LeanRidgelet.quadraticRelativeMeasure lam))

def LeanRidgelet.quadraticRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (hψ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticAnalysisKernel
        ψ)
      2
      ((LeanRidgelet.quadraticRelativeMeasure
            lam).prod
        MeasureTheory.volume)) :
  ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume) →L[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure
          lam))

Implementation after :=

:= E) ψ) 2
      ((quadraticRelativeMeasure lam).prod (volume : Measure E))) :
    Lp ℂ 2 (volume : Measure E) →L[ℂ] Lp ℂ 2 (quadraticRelativeMeasure lam) :=
  hilbertSchmidtKernelOperator hψ

The quadratic ridgelet transform: the analysis integral of the quadratic feature, as a bounded
operator from data `L²` to parameter `L²`.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.norm_quadraticMachine_le.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticSynthesisKernel σ) 2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure lam))) :
  ‖LeanRidgelet.quadraticMachine lam hσ‖ ≤
    (MeasureTheory.eLpNorm (LeanRidgelet.quadraticSynthesisKernel σ) 2
        (MeasureTheory.volume.prod
          (LeanRidgelet.quadraticRelativeMeasure lam))).toReal

theorem LeanRidgelet.norm_quadraticMachine_le.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticSynthesisKernel
        σ)
      2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure
          lam))) :
  ‖LeanRidgelet.quadraticMachine lam hσ‖ ≤
    (MeasureTheory.eLpNorm
        (LeanRidgelet.quadraticSynthesisKernel
          σ)
        2
        (MeasureTheory.volume.prod
          (LeanRidgelet.quadraticRelativeMeasure
            lam))).toReal

The operator norm of the quadratic machine is at most the `L²` norm of the feature.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.norm_quadraticRidgelet_le.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (hψ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticAnalysisKernel ψ) 2
      ((LeanRidgelet.quadraticRelativeMeasure lam).prod
        MeasureTheory.volume)) :
  ‖LeanRidgelet.quadraticRidgelet lam hψ‖ ≤
    (MeasureTheory.eLpNorm (LeanRidgelet.quadraticAnalysisKernel ψ) 2
        ((LeanRidgelet.quadraticRelativeMeasure lam).prod
          MeasureTheory.volume)).toReal

theorem LeanRidgelet.norm_quadraticRidgelet_le.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (hψ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticAnalysisKernel
        ψ)
      2
      ((LeanRidgelet.quadraticRelativeMeasure
            lam).prod
        MeasureTheory.volume)) :
  ‖LeanRidgelet.quadraticRidgelet lam
        hψ‖ ≤
    (MeasureTheory.eLpNorm
        (LeanRidgelet.quadraticAnalysisKernel
          ψ)
        2
        ((LeanRidgelet.quadraticRelativeMeasure
              lam).prod
          MeasureTheory.volume)).toReal

The operator norm of the quadratic ridgelet transform is at most the `L²` norm of the
feature.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.coeFn_quadraticMachine.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticSynthesisKernel σ) 2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure lam)))
  (γ :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure lam))) :
  ↑↑((LeanRidgelet.quadraticMachine lam hσ) γ) =ᵐ[MeasureTheory.volume]
    LeanRidgelet.bochnerSynthesis
      (LeanRidgelet.quadraticRelativeMeasure lam)
      (LeanRidgelet.quadraticVectorFeature σ) ↑↑γ

theorem LeanRidgelet.coeFn_quadraticMachine.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticSynthesisKernel
        σ)
      2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure
          lam)))
  (γ :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure
          lam))) :
  ↑↑((LeanRidgelet.quadraticMachine lam
            hσ)
          γ) =ᵐ[MeasureTheory.volume]
    LeanRidgelet.bochnerSynthesis
      (LeanRidgelet.quadraticRelativeMeasure
        lam)
      (LeanRidgelet.quadraticVectorFeature
        σ)
      ↑↑γ

The machine is the pointwise Bochner synthesis of the quadratic feature, almost everywhere.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.coeFn_quadraticRidgelet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (hψ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticAnalysisKernel ψ) 2
      ((LeanRidgelet.quadraticRelativeMeasure lam).prod
        MeasureTheory.volume))
  (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) :
  ↑↑((LeanRidgelet.quadraticRidgelet lam hψ)
          f) =ᵐ[LeanRidgelet.quadraticRelativeMeasure lam]
    LeanRidgelet.bochnerRidgelet MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f

theorem LeanRidgelet.coeFn_quadraticRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (hψ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticAnalysisKernel
        ψ)
      2
      ((LeanRidgelet.quadraticRelativeMeasure
            lam).prod
        MeasureTheory.volume))
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)) :
  ↑↑((LeanRidgelet.quadraticRidgelet lam
            hψ)
          f) =ᵐ[LeanRidgelet.quadraticRelativeMeasure
      lam]
    LeanRidgelet.bochnerRidgelet
      MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature
        ψ)
      ↑↑f

The ridgelet transform is the pointwise Bochner ridgelet of the quadratic feature, almost
everywhere.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.quadraticMachine_intertwines.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticSynthesisKernel σ) 2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure lam)))
  (g : E ≃ᵃ[ℝ] E)
  (γ :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure lam))) :
  (LeanRidgelet.quadraticMachine lam hσ)
      (↑((LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
              lam)
            g)
        γ) =
    ↑((LeanRidgelet.affineDataLpUnitaryRepresentation
            MeasureTheory.volume)
          g)
      ((LeanRidgelet.quadraticMachine lam hσ) γ)

theorem LeanRidgelet.quadraticMachine_intertwines.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticSynthesisKernel
        σ)
      2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure
          lam)))
  (g : E ≃ᵃ[ℝ] E)
  (γ :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure
          lam))) :
  (LeanRidgelet.quadraticMachine lam hσ)
      (↑((LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
              lam)
            g)
        γ) =
    ↑((LeanRidgelet.affineDataLpUnitaryRepresentation
            MeasureTheory.volume)
          g)
      ((LeanRidgelet.quadraticMachine lam
          hσ)
        γ)

The quadratic machine intertwines the quadratic parameter representation with the affine data
representation.  The identity holds in `L²`; the pointwise Bochner identity enters through the
almost-everywhere formulas for the two representations.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.quadraticRidgelet_intertwines.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (hψ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticAnalysisKernel ψ) 2
      ((LeanRidgelet.quadraticRelativeMeasure lam).prod
        MeasureTheory.volume))
  (g : E ≃ᵃ[ℝ] E) (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) :
  (LeanRidgelet.quadraticRidgelet lam hψ)
      (↑((LeanRidgelet.affineDataLpUnitaryRepresentation
              MeasureTheory.volume)
            g)
        f) =
    ↑((LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
            lam)
          g)
      ((LeanRidgelet.quadraticRidgelet lam hψ) f)

theorem LeanRidgelet.quadraticRidgelet_intertwines.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (hψ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticAnalysisKernel
        ψ)
      2
      ((LeanRidgelet.quadraticRelativeMeasure
            lam).prod
        MeasureTheory.volume))
  (g : E ≃ᵃ[ℝ] E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)) :
  (LeanRidgelet.quadraticRidgelet lam hψ)
      (↑((LeanRidgelet.affineDataLpUnitaryRepresentation
              MeasureTheory.volume)
            g)
        f) =
    ↑((LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
            lam)
          g)
      ((LeanRidgelet.quadraticRidgelet lam
          hψ)
        f)

The quadratic ridgelet transform intertwines the affine data representation with the quadratic
parameter representation.

defdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

def LeanRidgelet.quadraticEquivariantMachine.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticSynthesisKernel σ) 2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure lam))) :
  LeanRidgelet.JointEquivariantMachine
    (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
        lam).toContRepresentation
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation

def LeanRidgelet.quadraticEquivariantMachine.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticSynthesisKernel
        σ)
      2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure
          lam))) :
  LeanRidgelet.JointEquivariantMachine
    (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
        lam).toContRepresentation
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation

Implementation after :=

:= E) σ) 2
      ((volume : Measure E).prod (quadraticRelativeMeasure lam))) :
    JointEquivariantMachine
      (quadraticRelativeParameterLpUnitaryRepresentation lam).toContRepresentation
      (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E)).toContRepresentation where
  __ := quadraticMachine lam hσ
  isIntertwining' g :=
    ContinuousLinearMap.ext fun γ ↦ quadraticMachine_intertwines lam hσ g γ

The quadratic machine as an intertwining map, hence as a joint-equivariant machine in the sense
of the reconstruction theorem.

defdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

def LeanRidgelet.quadraticEquivariantRidgelet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (hψ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticAnalysisKernel ψ) 2
      ((LeanRidgelet.quadraticRelativeMeasure lam).prod
        MeasureTheory.volume)) :
  LeanRidgelet.JointEquivariantRidgelet
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation
    (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
        lam).toContRepresentation

def LeanRidgelet.quadraticEquivariantRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (hψ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticAnalysisKernel
        ψ)
      2
      ((LeanRidgelet.quadraticRelativeMeasure
            lam).prod
        MeasureTheory.volume)) :
  LeanRidgelet.JointEquivariantRidgelet
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation
    (LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
        lam).toContRepresentation

Implementation after :=

:= E) ψ) 2
      ((quadraticRelativeMeasure lam).prod (volume : Measure E))) :
    JointEquivariantRidgelet
      (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E)).toContRepresentation
      (quadraticRelativeParameterLpUnitaryRepresentation lam).toContRepresentation where
  __ := quadraticRidgelet lam hψ
  isIntertwining' g :=
    ContinuousLinearMap.ext fun f ↦ quadraticRidgelet_intertwines lam hψ g f

The quadratic ridgelet transform as an intertwining map.

theoremdefined in LeanRidgelet/HA/QuadraticBounded.lean

complete

theorem LeanRidgelet.quadratic_reconstruction_of_memLp.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticSynthesisKernel σ) 2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure lam)))
  (hψ :
    MeasureTheory.MemLp (LeanRidgelet.quadraticAnalysisKernel ψ) 2
      ((LeanRidgelet.quadraticRelativeMeasure lam).prod
        MeasureTheory.volume)) :
  ∃ c,
    LeanRidgelet.jointReconstructionOperator
          (LeanRidgelet.quadraticEquivariantMachine lam hσ)
          (LeanRidgelet.quadraticEquivariantRidgelet lam hψ) =
        c •
          ContinuousLinearMap.id ℂ
            ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) ∧
      (c ≠ 0 →
        Function.RightInverse
          ⇑(c⁻¹ •
              (LeanRidgelet.quadraticEquivariantRidgelet lam
                  hψ).toContinuousLinearMap)
          ⇑(LeanRidgelet.quadraticEquivariantMachine lam hσ))

theorem LeanRidgelet.quadratic_reconstruction_of_memLp.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (hσ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticSynthesisKernel
        σ)
      2
      (MeasureTheory.volume.prod
        (LeanRidgelet.quadraticRelativeMeasure
          lam)))
  (hψ :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticAnalysisKernel
        ψ)
      2
      ((LeanRidgelet.quadraticRelativeMeasure
            lam).prod
        MeasureTheory.volume)) :
  ∃ c,
    LeanRidgelet.jointReconstructionOperator
          (LeanRidgelet.quadraticEquivariantMachine
            lam hσ)
          (LeanRidgelet.quadraticEquivariantRidgelet
            lam hψ) =
        c •
          ContinuousLinearMap.id ℂ
            ↥(MeasureTheory.Lp ℂ 2
                MeasureTheory.volume) ∧
      (c ≠ 0 →
        Function.RightInverse
          ⇑(c⁻¹ •
              (LeanRidgelet.quadraticEquivariantRidgelet
                  lam
                  hψ).toContinuousLinearMap)
          ⇑(LeanRidgelet.quadraticEquivariantMachine
              lam hσ))

**Section 7 with no boundedness hypothesis left.**  If the quadratic feature is square
integrable for the product of the data measure with the relatively invariant parameter measure, then
the reconstruction operator of the quadratic-form network is a scalar multiple of the identity, and
a nonzero scalar makes the normalized quadratic ridgelet transform a right inverse of the quadratic
machine.  Square integrability of the feature is the analytic input of the article's boundedness
appendix; everything else is proved.

The scalarity is Schur's lemma applied to the affine data representation, and its topological irreducibility is proved in this development rather than assumed, so the Section 7 conclusion rests on no placeholder. The parameter side is the relatively invariant model of the previous node, which is what makes the two representations balanced; with the additive Haar parameter measure they are not, and no such pair of intertwiners exists. Boundedness is no longer assumed. Reading the feature as an integral kernel, the Hilbert-Schmidt bound of the Mathlib candidate layer bundles each of the two integrals into a continuous linear map on L² whose operator norm is at most the L² norm of the feature, and the quasi-invariant Bochner identities upgrade them to intertwining maps. What is assumed instead is exactly the analytic input the article isolates in its boundedness appendix: that the feature is square integrable for the product of the data measure with the relatively invariant parameter measure. For a general activation that is false, so it cannot be proved.

The intertwining is proved as an equality of L² classes rather than pointwise, which is the level at which both the operators and the representations are defined: a class is an almost-everywhere equivalence class, and the pointwise Bochner identities enter through the almost-everywhere formulas for the two representations. Two small congruence facts are what make that work — each action respects almost-everywhere equality of representatives, because the inverse action is quasi measure preserving for the corresponding measure.

Theorem4.5.6

Statement uses 2

Statement dependency previews

Preview

Theorem 4.5.5

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

used by 1✓L∃∀N

The composite kernel, and why it is not the route to universality. The reconstruction operator is bounded as soon as the composite kernel of the pair of features is square integrable, without either integral operator being bounded — but then the reconstruction constant is forced to vanish.

Lean code for Theorem4.5.6●8 declarations

defdefined in LeanRidgelet/HA/QuadraticComposite.lean

complete

def LeanRidgelet.quadraticCompositeKernel.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (σ ψ : ℝ → ℂ) : E × E → ℂ

def LeanRidgelet.quadraticCompositeKernel.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (σ ψ : ℝ → ℂ) : E × E → ℂ

Implementation after :=

:=
  fun p ↦ ∫ ξ, quadraticVectorFeature ψ p.2 ξ * conj (quadraticVectorFeature σ p.1 ξ)
    ∂quadraticRelativeMeasure lam

The composite kernel of an analysis feature `ψ` against a synthesis feature `σ`, integrated over
the relatively invariant parameter measure.  This is the kernel of the reconstruction operator.

defdefined in LeanRidgelet/HA/QuadraticComposite.lean

complete

def LeanRidgelet.quadraticCompositeOperator.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp (LeanRidgelet.quadraticCompositeKernel lam σ ψ)
      2 (MeasureTheory.volume.prod MeasureTheory.volume)) :
  ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) →L[ℂ]
    ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)

def LeanRidgelet.quadraticCompositeOperator.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticCompositeKernel
        lam σ ψ)
      2
      (MeasureTheory.volume.prod
        MeasureTheory.volume)) :
  ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume) →L[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)

Implementation after :=

:=
  hilbertSchmidtKernelOperator hk

The reconstruction operator of the quadratic-form network, as a bounded operator on data `L²`,
whenever its composite kernel is square integrable.

theoremdefined in LeanRidgelet/HA/QuadraticComposite.lean

complete

theorem LeanRidgelet.norm_quadraticCompositeOperator_le.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp (LeanRidgelet.quadraticCompositeKernel lam σ ψ)
      2 (MeasureTheory.volume.prod MeasureTheory.volume)) :
  ‖LeanRidgelet.quadraticCompositeOperator lam hk‖ ≤
    (MeasureTheory.eLpNorm
        (LeanRidgelet.quadraticCompositeKernel lam σ ψ) 2
        (MeasureTheory.volume.prod MeasureTheory.volume)).toReal

theorem LeanRidgelet.norm_quadraticCompositeOperator_le.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticCompositeKernel
        lam σ ψ)
      2
      (MeasureTheory.volume.prod
        MeasureTheory.volume)) :
  ‖LeanRidgelet.quadraticCompositeOperator
        lam hk‖ ≤
    (MeasureTheory.eLpNorm
        (LeanRidgelet.quadraticCompositeKernel
          lam σ ψ)
        2
        (MeasureTheory.volume.prod
          MeasureTheory.volume)).toReal

The operator norm of the reconstruction operator is at most the `L²` norm of the composite
kernel.  This is condition T1 of the article's boundedness appendix.

theoremdefined in LeanRidgelet/HA/QuadraticComposite.lean

complete

theorem LeanRidgelet.coeFn_quadraticCompositeOperator.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp (LeanRidgelet.quadraticCompositeKernel lam σ ψ)
      2 (MeasureTheory.volume.prod MeasureTheory.volume))
  (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume))
  (hfub :
    ∀ᵐ (x : E),
      MeasureTheory.Integrable
        (Function.uncurry fun ξ y ↦
          ↑↑f y *
              (starRingEnd ℂ)
                (LeanRidgelet.quadraticVectorFeature ψ y ξ) *
            LeanRidgelet.quadraticVectorFeature σ x ξ)
        ((LeanRidgelet.quadraticRelativeMeasure lam).prod
          MeasureTheory.volume)) :
  ↑↑((LeanRidgelet.quadraticCompositeOperator lam hk)
          f) =ᵐ[MeasureTheory.volume]
    LeanRidgelet.bochnerSynthesis
      (LeanRidgelet.quadraticRelativeMeasure lam)
      (LeanRidgelet.quadraticVectorFeature σ)
      (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f)

theorem LeanRidgelet.coeFn_quadraticCompositeOperator.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticCompositeKernel
        lam σ ψ)
      2
      (MeasureTheory.volume.prod
        MeasureTheory.volume))
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume))
  (hfub :
    ∀ᵐ (x : E),
      MeasureTheory.Integrable
        (Function.uncurry fun ξ y ↦
          ↑↑f y *
              (starRingEnd ℂ)
                (LeanRidgelet.quadraticVectorFeature
                  ψ y ξ) *
            LeanRidgelet.quadraticVectorFeature
              σ x ξ)
        ((LeanRidgelet.quadraticRelativeMeasure
              lam).prod
          MeasureTheory.volume)) :
  ↑↑((LeanRidgelet.quadraticCompositeOperator
            lam hk)
          f) =ᵐ[MeasureTheory.volume]
    LeanRidgelet.bochnerSynthesis
      (LeanRidgelet.quadraticRelativeMeasure
        lam)
      (LeanRidgelet.quadraticVectorFeature
        σ)
      (LeanRidgelet.bochnerRidgelet
        MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature
          ψ)
        ↑↑f)

The kernel operator of the composite kernel is the pointwise composite of the Bochner synthesis
with the Bochner ridgelet.  The exchange of the two integrals is the Fubini hypothesis.

theoremdefined in LeanRidgelet/HA/QuadraticComposite.lean

complete

theorem LeanRidgelet.bochnerRidgelet_quadraticVectorFeature_congr_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (ψ : ℝ → ℂ)
  {F₁ F₂ : E → ℂ} (hF : F₁ =ᵐ[MeasureTheory.volume] F₂) :
  LeanRidgelet.bochnerRidgelet MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature ψ) F₁ =
    LeanRidgelet.bochnerRidgelet MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature ψ) F₂

theorem LeanRidgelet.bochnerRidgelet_quadraticVectorFeature_congr_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (ψ : ℝ → ℂ) {F₁ F₂ : E → ℂ}
  (hF : F₁ =ᵐ[MeasureTheory.volume] F₂) :
  LeanRidgelet.bochnerRidgelet
      MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature
        ψ)
      F₁ =
    LeanRidgelet.bochnerRidgelet
      MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature
        ψ)
      F₂

The Bochner ridgelet of the quadratic feature only depends on the data function up to almost
everywhere equality.

theoremdefined in LeanRidgelet/HA/QuadraticComposite.lean

complete

theorem LeanRidgelet.quadraticCompositeOperator_intertwines.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp (LeanRidgelet.quadraticCompositeKernel lam σ ψ)
      2 (MeasureTheory.volume.prod MeasureTheory.volume))
  (hfub :
    ∀ (h : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      ∀ᵐ (x : E),
        MeasureTheory.Integrable
          (Function.uncurry fun ξ y ↦
            ↑↑h y *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature ψ y ξ) *
              LeanRidgelet.quadraticVectorFeature σ x ξ)
          ((LeanRidgelet.quadraticRelativeMeasure lam).prod
            MeasureTheory.volume))
  (g : E ≃ᵃ[ℝ] E) (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) :
  (LeanRidgelet.quadraticCompositeOperator lam hk)
      (↑((LeanRidgelet.affineDataLpUnitaryRepresentation
              MeasureTheory.volume)
            g)
        f) =
    ↑((LeanRidgelet.affineDataLpUnitaryRepresentation
            MeasureTheory.volume)
          g)
      ((LeanRidgelet.quadraticCompositeOperator lam hk) f)

theorem LeanRidgelet.quadraticCompositeOperator_intertwines.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticCompositeKernel
        lam σ ψ)
      2
      (MeasureTheory.volume.prod
        MeasureTheory.volume))
  (hfub :
    ∀
      (h :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      ∀ᵐ (x : E),
        MeasureTheory.Integrable
          (Function.uncurry fun ξ y ↦
            ↑↑h y *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature
                    ψ y ξ) *
              LeanRidgelet.quadraticVectorFeature
                σ x ξ)
          ((LeanRidgelet.quadraticRelativeMeasure
                lam).prod
            MeasureTheory.volume))
  (g : E ≃ᵃ[ℝ] E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)) :
  (LeanRidgelet.quadraticCompositeOperator
        lam hk)
      (↑((LeanRidgelet.affineDataLpUnitaryRepresentation
              MeasureTheory.volume)
            g)
        f) =
    ↑((LeanRidgelet.affineDataLpUnitaryRepresentation
            MeasureTheory.volume)
          g)
      ((LeanRidgelet.quadraticCompositeOperator
          lam hk)
        f)

The reconstruction operator intertwines the affine data representation with itself.  The
identity holds in `L²`, and both pointwise Bochner identities enter through the almost-everywhere
formula for the data representation.

defdefined in LeanRidgelet/HA/QuadraticComposite.lean

complete

def LeanRidgelet.quadraticCompositeIntertwiner.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp (LeanRidgelet.quadraticCompositeKernel lam σ ψ)
      2 (MeasureTheory.volume.prod MeasureTheory.volume))
  (hfub :
    ∀ (h : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      ∀ᵐ (x : E),
        MeasureTheory.Integrable
          (Function.uncurry fun ξ y ↦
            ↑↑h y *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature ψ y ξ) *
              LeanRidgelet.quadraticVectorFeature σ x ξ)
          ((LeanRidgelet.quadraticRelativeMeasure lam).prod
            MeasureTheory.volume)) :
  LeanRidgelet.JointEquivariantMachine
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation

def LeanRidgelet.quadraticCompositeIntertwiner.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticCompositeKernel
        lam σ ψ)
      2
      (MeasureTheory.volume.prod
        MeasureTheory.volume))
  (hfub :
    ∀
      (h :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      ∀ᵐ (x : E),
        MeasureTheory.Integrable
          (Function.uncurry fun ξ y ↦
            ↑↑h y *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature
                    ψ y ξ) *
              LeanRidgelet.quadraticVectorFeature
                σ x ξ)
          ((LeanRidgelet.quadraticRelativeMeasure
                lam).prod
            MeasureTheory.volume)) :
  LeanRidgelet.JointEquivariantMachine
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation

Implementation after :=

:= ℂ) (volume : Measure E)).toContRepresentation
      (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E)).toContRepresentation where
  __ := quadraticCompositeOperator lam hk
  isIntertwining' g :=
    ContinuousLinearMap.ext fun f ↦ quadraticCompositeOperator_intertwines lam hk hfub g f

The reconstruction operator as a bundled intertwining endomorphism of the data
representation.

theoremdefined in LeanRidgelet/HA/QuadraticComposite.lean

complete

theorem LeanRidgelet.quadraticComposite_reconstruction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp (LeanRidgelet.quadraticCompositeKernel lam σ ψ)
      2 (MeasureTheory.volume.prod MeasureTheory.volume))
  (hfub :
    ∀ (h : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      ∀ᵐ (x : E),
        MeasureTheory.Integrable
          (Function.uncurry fun ξ y ↦
            ↑↑h y *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature ψ y ξ) *
              LeanRidgelet.quadraticVectorFeature σ x ξ)
          ((LeanRidgelet.quadraticRelativeMeasure lam).prod
            MeasureTheory.volume)) :
  ∃ c,
    (LeanRidgelet.quadraticCompositeIntertwiner lam hk
          hfub).toContinuousLinearMap =
      c •
        ContinuousLinearMap.id ℂ
          ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)

theorem LeanRidgelet.quadraticComposite_reconstruction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (hk :
    MeasureTheory.MemLp
      (LeanRidgelet.quadraticCompositeKernel
        lam σ ψ)
      2
      (MeasureTheory.volume.prod
        MeasureTheory.volume))
  (hfub :
    ∀
      (h :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      ∀ᵐ (x : E),
        MeasureTheory.Integrable
          (Function.uncurry fun ξ y ↦
            ↑↑h y *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature
                    ψ y ξ) *
              LeanRidgelet.quadraticVectorFeature
                σ x ξ)
          ((LeanRidgelet.quadraticRelativeMeasure
                lam).prod
            MeasureTheory.volume)) :
  ∃ c,
    (LeanRidgelet.quadraticCompositeIntertwiner
          lam hk
          hfub).toContinuousLinearMap =
      c •
        ContinuousLinearMap.id ℂ
          ↥(MeasureTheory.Lp ℂ 2
              MeasureTheory.volume)

**Section 7 for an unbounded activation.**  If the composite kernel of the pair of features is
square integrable and the two integrals may be exchanged, then the reconstruction operator of the
quadratic-form network is a scalar multiple of the identity.  Neither the machine nor the ridgelet
transform is assumed bounded, which is what makes the statement available for an activation of
polynomial growth such as the rectified linear unit.

Square integrability of the feature itself fails for every activation of polynomial growth, the rectified linear unit included: at a fixed parameter the integral of the squared feature over the data space already diverges. The article's other sufficient condition asks less, only that the composite kernel — the analysis feature paired against the synthesis feature and integrated over the parameter measure — be square integrable, and that is not automatically false. Fubini identifies the operator of that kernel with the pointwise composite of the Bochner synthesis and the Bochner ridgelet, and the two quasi-invariant identities make it an intertwining endomorphism of the data representation, so the reconstruction theorem applies with no boundedness hypothesis on either half.

The conclusion, however, is that the constant is zero. An operator with a square-integrable kernel is Hilbert-Schmidt, hence compact, while a scalar operator on an infinite-dimensional space is compact only for the zero scalar. The same argument applies a fortiori to square integrability of the feature, which makes both halves Hilbert-Schmidt. So neither condition can produce the nonzero constant universality needs, and that is why the L1 and L2 theories bound the two halves separately through a weighted intermediate space instead — a weighted parameter measure in one, transported coordinates with a weighted activation space in the other — where neither operator is Hilbert-Schmidt and the composite is bounded without being compact. For the quadratic feature that remains new analysis, and the weight has to go on the coefficient space, since the balance pins the parameter measure down.

Lemma4.5.7

uses 1

Used by 4

Reverse dependency previews

Preview

Lemma 4.5.10

Loading preview

Reverse dependency preview content is loaded from the rendered-fragment cache.

✓L∃∀N

Transferring derivatives across the analysis transform. The constant coefficient of a quadratic parameter enters the feature additively, so differentiating the analysis transform in it replaces the analysis feature by its derivative and changes nothing else.

Lean code for Lemma4.5.7●5 theorems

theoremdefined in LeanRidgelet/HA/QuadraticTransfer.lean

complete

theorem LeanRidgelet.quadraticArgument_const_add.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (x : E) (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (c : ℝ) :
  LeanRidgelet.quadraticArgument x (A, b, c) =
    inner ℝ x (↑A x) + inner ℝ x b + c

theorem LeanRidgelet.quadraticArgument_const_add.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (x : E)
  (A : LeanRidgelet.QuadraticSymmetric E)
  (b : E) (c : ℝ) :
  LeanRidgelet.quadraticArgument x
      (A, b, c) =
    inner ℝ x (↑A x) + inner ℝ x b + c

The constant coefficient of a quadratic parameter enters the feature additively.

theoremdefined in LeanRidgelet/HA/QuadraticTransfer.lean

complete

theorem LeanRidgelet.hasDerivAt_quadraticAnalysisPairing.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {ρ ρ' : ℝ → ℂ} (hderiv : ∀ (z : ℝ), HasDerivAt ρ (ρ' z) z) (v : ℂ)
  (x : E) (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (c : ℝ) :
  HasDerivAt
    (fun c ↦
      v *
        (starRingEnd ℂ)
          (LeanRidgelet.quadraticVectorFeature ρ x (A, b, c)))
    (v *
      (starRingEnd ℂ)
        (LeanRidgelet.quadraticVectorFeature ρ' x (A, b, c)))
    c

theorem LeanRidgelet.hasDerivAt_quadraticAnalysisPairing.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {ρ ρ' : ℝ → ℂ}
  (hderiv :
    ∀ (z : ℝ), HasDerivAt ρ (ρ' z) z)
  (v : ℂ) (x : E)
  (A : LeanRidgelet.QuadraticSymmetric E)
  (b : E) (c : ℝ) :
  HasDerivAt
    (fun c ↦
      v *
        (starRingEnd ℂ)
          (LeanRidgelet.quadraticVectorFeature
            ρ x (A, b, c)))
    (v *
      (starRingEnd ℂ)
        (LeanRidgelet.quadraticVectorFeature
          ρ' x (A, b, c)))
    c

Differentiating the integrand of the analysis transform in the constant coefficient replaces the
analysis feature by its derivative.  This is the transfer of a derivative across the convolution in
the additive parameter.

theoremdefined in LeanRidgelet/HA/QuadraticTransfer.lean

complete

theorem LeanRidgelet.hasDerivAt_bochnerRidgelet_quadraticVectorFeature.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  {ρ ρ' : ℝ → ℂ} (f : E → ℂ) (A : LeanRidgelet.QuadraticSymmetric E)
  (b : E) (c₀ : ℝ) {ε : ℝ} (hε : 0 < ε) {bound : E → ℝ}
  (hderiv : ∀ (z : ℝ), HasDerivAt ρ (ρ' z) z)
  (hmeas :
    ∀ᶠ (c : ℝ) in nhds c₀,
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature ρ x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    MeasureTheory.Integrable
      (fun x ↦
        f x *
          (starRingEnd ℂ)
            (LeanRidgelet.quadraticVectorFeature ρ x (A, b, c₀)))
      MeasureTheory.volume)
  (hmeas' :
    MeasureTheory.AEStronglyMeasurable
      (fun x ↦
        f x *
          (starRingEnd ℂ)
            (LeanRidgelet.quadraticVectorFeature ρ' x (A, b, c₀)))
      MeasureTheory.volume)
  (hbound :
    ∀ᵐ (x : E),
      ∀ c ∈ Metric.ball c₀ ε,
        ‖f x *
              (starRingEnd ℂ)
                (LeanRidgelet.quadraticVectorFeature ρ' x (A, b, c))‖ ≤
          bound x)
  (hboundint : MeasureTheory.Integrable bound MeasureTheory.volume) :
  HasDerivAt
    (fun c ↦
      LeanRidgelet.bochnerRidgelet MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature ρ) f (A, b, c))
    (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature ρ') f (A, b, c₀))
    c₀

theorem LeanRidgelet.hasDerivAt_bochnerRidgelet_quadraticVectorFeature.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {ρ ρ' : ℝ → ℂ} (f : E → ℂ)
  (A : LeanRidgelet.QuadraticSymmetric E)
  (b : E) (c₀ : ℝ) {ε : ℝ} (hε : 0 < ε)
  {bound : E → ℝ}
  (hderiv :
    ∀ (z : ℝ), HasDerivAt ρ (ρ' z) z)
  (hmeas :
    ∀ᶠ (c : ℝ) in nhds c₀,
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                ρ x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    MeasureTheory.Integrable
      (fun x ↦
        f x *
          (starRingEnd ℂ)
            (LeanRidgelet.quadraticVectorFeature
              ρ x (A, b, c₀)))
      MeasureTheory.volume)
  (hmeas' :
    MeasureTheory.AEStronglyMeasurable
      (fun x ↦
        f x *
          (starRingEnd ℂ)
            (LeanRidgelet.quadraticVectorFeature
              ρ' x (A, b, c₀)))
      MeasureTheory.volume)
  (hbound :
    ∀ᵐ (x : E),
      ∀ c ∈ Metric.ball c₀ ε,
        ‖f x *
              (starRingEnd ℂ)
                (LeanRidgelet.quadraticVectorFeature
                  ρ' x (A, b, c))‖ ≤
          bound x)
  (hboundint :
    MeasureTheory.Integrable bound
      MeasureTheory.volume) :
  HasDerivAt
    (fun c ↦
      LeanRidgelet.bochnerRidgelet
        MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature
          ρ)
        f (A, b, c))
    (LeanRidgelet.bochnerRidgelet
      MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature
        ρ')
      f (A, b, c₀))
    c₀

The analysis transform of the quadratic feature is differentiable in the constant coefficient of
the parameter, with derivative the analysis transform of the derivative feature.  The hypotheses are
those of the dominated-convergence differentiation lemma: the dominating function is what smoothness
of the data and of the analysis feature supplies.

theoremdefined in LeanRidgelet/HA/QuadraticTransfer.lean

complete

theorem LeanRidgelet.iteratedDeriv_bochnerRidgelet_quadraticVectorFeature.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ) (A : LeanRidgelet.QuadraticSymmetric E)
  (b : E) {bound : ℕ → E → ℝ}
  (hderiv : ∀ (i : ℕ) (z : ℝ), HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀ (i : ℕ) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀ (i : ℕ) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀ (i : ℕ),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound i x)
  (hboundint :
    ∀ (i : ℕ), MeasureTheory.Integrable (bound i) MeasureTheory.volume)
  (k : ℕ) :
  (iteratedDeriv k fun c ↦
      LeanRidgelet.bochnerRidgelet MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature (ρ 0)) f (A, b, c)) =
    fun c ↦
    LeanRidgelet.bochnerRidgelet MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature (ρ k)) f (A, b, c)

theorem LeanRidgelet.iteratedDeriv_bochnerRidgelet_quadraticVectorFeature.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ)
  (A : LeanRidgelet.QuadraticSymmetric E)
  (b : E) {bound : ℕ → E → ℝ}
  (hderiv :
    ∀ (i : ℕ) (z : ℝ),
      HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀ (i : ℕ) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀ (i : ℕ) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀ (i : ℕ),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature
                    (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound i x)
  (hboundint :
    ∀ (i : ℕ),
      MeasureTheory.Integrable (bound i)
        MeasureTheory.volume)
  (k : ℕ) :
  (iteratedDeriv k fun c ↦
      LeanRidgelet.bochnerRidgelet
        MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature
          (ρ 0))
        f (A, b, c)) =
    fun c ↦
    LeanRidgelet.bochnerRidgelet
      MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature
        (ρ k))
      f (A, b, c)

**Iterating the transfer.**  Given a sequence of analysis features each the derivative of the
previous one, the `k`-th derivative of the analysis transform in the constant coefficient is the
analysis transform of the `k`-th feature.  This is the identity that produces the frequency weight:
on the Fourier side in the constant coefficient the left-hand side is multiplication by the `k`-th
power of the frequency, so the transform lies in the space weighted by the `2k`-th power of the
frequency with its norm controlled by the `k`-th feature.

theoremdefined in LeanRidgelet/ToMathlib/HilbertSchmidtKernel.lean

complete

theorem MeasureTheory.eLpNorm_integral_weighted_le.{u_1, u_2} {α : Type u_1}
  {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β]
  {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β}
  [MeasureTheory.SFinite ν] {w : β → ℝ} (hw : ∀ (y : β), 0 < w y)
  {φ : α × β → ℂ} {γ : β → ℂ} {K : α × β → ℂ} {Γ : β → ℂ}
  (hKdef : ∀ (p : α × β), K p = φ p / ↑√(w p.2))
  (hΓdef : ∀ (y : β), Γ y = γ y * ↑√(w y))
  (hK : MeasureTheory.MemLp K 2 (μ.prod ν))
  (hΓ : MeasureTheory.MemLp Γ 2 ν) :
  MeasureTheory.eLpNorm
      (fun x ↦ ∫ (y : β), γ y * (starRingEnd ℂ) (φ (x, y)) ∂ν) 2 μ ≤
    MeasureTheory.eLpNorm K 2 (μ.prod ν) * MeasureTheory.eLpNorm Γ 2 ν

theorem MeasureTheory.eLpNorm_integral_weighted_le.{u_1,
    u_2}
  {α : Type u_1} {β : Type u_2}
  [MeasurableSpace α] [MeasurableSpace β]
  {μ : MeasureTheory.Measure α}
  {ν : MeasureTheory.Measure β}
  [MeasureTheory.SFinite ν] {w : β → ℝ}
  (hw : ∀ (y : β), 0 < w y)
  {φ : α × β → ℂ} {γ : β → ℂ}
  {K : α × β → ℂ} {Γ : β → ℂ}
  (hKdef :
    ∀ (p : α × β), K p = φ p / ↑√(w p.2))
  (hΓdef : ∀ (y : β), Γ y = γ y * ↑√(w y))
  (hK :
    MeasureTheory.MemLp K 2 (μ.prod ν))
  (hΓ : MeasureTheory.MemLp Γ 2 ν) :
  MeasureTheory.eLpNorm
      (fun x ↦
        ∫ (y : β),
          γ y *
            (starRingEnd ℂ) (φ (x, y)) ∂ν)
      2 μ ≤
    MeasureTheory.eLpNorm K 2 (μ.prod ν) *
      MeasureTheory.eLpNorm Γ 2 ν

**Synthesis from a weighted coefficient space.**  Splitting a positive weight between the
coefficient and the feature turns the kernel estimate into a bound from the weighted `L²` of
coefficients: the operator is controlled by the `L²` norm of the feature divided by the square root
of the weight, against the `L²` norm of the coefficient multiplied by it.  This is the pattern by
which the L1 theory bounds its transforms against a weighted parameter measure.

Note what the hypothesis says.  Finiteness of the weighted feature norm is square integrability of
the feature divided by the square root of the weight, so the operator is again Hilbert--Schmidt, and
a weight cannot by itself escape that: any bound obtained from Cauchy--Schwarz pointwise in the data
variable carries an `L²` majorant and hence a square-integrable kernel.  Escaping compactness
requires orthogonality in the parameter variable, which is what a Plancherel argument supplies and
this estimate does not.

This is the step that escapes the compactness obstruction of the previous node. The analysis transform is a convolution in the constant coefficient, so a derivative in that coefficient can be moved onto the analysis feature; iterating gives, for a k-times differentiable analysis feature, that the k-th derivative of the transform is the transform of the k-th derivative feature. That iteration is proved, for a sequence of features each the derivative of the previous one. On the Fourier side in that variable it is multiplication by the k-th power of the frequency, so the transform lies in the space weighted by the 2k-th power of the frequency, and the size of its norm there is controlled by the k-th feature together with the data. That exponent is the analogue of the smoothness index of the weighted activation spaces of the L2 theory; the growth index arises the same way, from multiplication by a polynomial in the constant coefficient, which on the Fourier side is a derivative. Neither weight can be moved onto the parameter measure, whose balance is already fixed, which is why they belong on the coefficient space.

The weighted estimate itself is recorded in the Mathlib candidate layer, together with the warning that a weight split between coefficient and feature by Cauchy--Schwarz is still a square-integrable kernel: the escape comes from the frequency weight and the orthogonality behind it, not from reweighting alone.

Joining the iterated transfer to the one-variable weighted Sobolev identity gives the estimate for each slice of the analysis transform in the constant coefficient: the frequency-weighted norm of the slice's angular Fourier transform is the Plancherel constant times the norm of the slice of the transform of the k-th feature. That is proved, for frozen values of the other two coefficients, with the hypotheses of both inputs threaded through and stated on the features rather than on the derivatives of the slice.

Integrating the slices over the remaining coefficients does not follow, and the reason is worth recording. The restriction and the density both factor through the symmetric coefficient alone, so they are not the problem; but the parameter measure is an abstract additive Haar measure on a triple product, which is not presented as a product over the constant coefficient, and after the Fourier transform the object lives over a space whose last coordinate is a frequency carrying Lebesgue measure rather than the restricted weighted measure. Beyond that bookkeeping, the exchange needs joint measurability of the transform in the parameter, and that is now available: both Bochner transforms are strongly measurable in the variable they produce, given a jointly measurable feature and an s-finite measure to integrate against, and for the quadratic feature the joint hypothesis is discharged by joint continuity of its scalar argument — evaluation of a continuous linear map is continuous in the pair, so the argument is continuous in data and parameter together. The measure bookkeeping is settled too: the degenerate locus and the weight read only the symmetric coefficient, so under the associativity transport the parameter measure becomes a positive multiple of a product over the constant coefficient, and an integral against it is the iterated integral that a Fubini step wants. So the measure poses no obstruction. What still separates the slice estimate from an integrated one is that the weighted side is a function of a frequency rather than of a constant coefficient, so there are not two functions on the same space to integrate — the remaining work is to say what the integrated statement should be, not to push a known one through.

Lemma4.5.8

Statement uses 2

Statement dependency previews

Preview

Lemma 4.2.6

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

used by 0✓L∃∀N

Measurability of the transforms in the variable they produce. Both Bochner transforms are strongly measurable, given a jointly measurable feature and an s-finite measure to integrate against; for the quadratic feature the joint hypothesis follows from joint continuity of its scalar argument.

Lean code for Lemma4.5.8●11 theorems

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.aestronglyMeasurable_bochnerRidgelet.{u_1, u_2, u_3}
  {X : Type u_1} {Ξ : Type u_2} {Y : Type u_3} [MeasurableSpace X]
  [MeasurableSpace Ξ] [NormedAddCommGroup Y] [InnerProductSpace ℂ Y]
  (μ : MeasureTheory.Measure X) [MeasureTheory.SFinite μ]
  (ν : MeasureTheory.Measure Ξ) [MeasureTheory.SFinite ν]
  {ψ : X → Ξ → Y} {f : X → Y}
  (hψ :
    MeasureTheory.AEStronglyMeasurable (Function.uncurry ψ) (μ.prod ν))
  (hf : MeasureTheory.AEStronglyMeasurable f μ) :
  MeasureTheory.AEStronglyMeasurable
    (LeanRidgelet.bochnerRidgelet μ ψ f) ν

theorem LeanRidgelet.aestronglyMeasurable_bochnerRidgelet.{u_1,
    u_2, u_3}
  {X : Type u_1} {Ξ : Type u_2}
  {Y : Type u_3} [MeasurableSpace X]
  [MeasurableSpace Ξ]
  [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  (μ : MeasureTheory.Measure X)
  [MeasureTheory.SFinite μ]
  (ν : MeasureTheory.Measure Ξ)
  [MeasureTheory.SFinite ν]
  {ψ : X → Ξ → Y} {f : X → Y}
  (hψ :
    MeasureTheory.AEStronglyMeasurable
      (Function.uncurry ψ) (μ.prod ν))
  (hf :
    MeasureTheory.AEStronglyMeasurable f
      μ) :
  MeasureTheory.AEStronglyMeasurable
    (LeanRidgelet.bochnerRidgelet μ ψ f) ν

**The analysis transform is measurable in the parameter.**  If the feature is jointly almost
everywhere strongly measurable for the product of the data measure with the parameter measure and
the data is measurable, then `LeanRidgelet.bochnerRidgelet` is almost everywhere strongly
measurable on the parameter space.

The integrand `(ξ, x) ↦ ⟪ψ x ξ, f x⟫` is jointly measurable, being the continuous inner product of
two jointly measurable functions, so
`MeasureTheory.AEStronglyMeasurable.integral_prod_right'` applies.  The hypothesis on the feature is
stated in the `X × Ξ` order, which is the uncurried order of `ψ : X → Ξ → Y`; the Fubini lemma wants
the parameter outside, so the proof exchanges the two arguments, and that exchange is what makes
σ-finiteness of the parameter measure a hypothesis here.

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.stronglyMeasurable_bochnerRidgelet.{u_1, u_2, u_3}
  {X : Type u_1} {Ξ : Type u_2} {Y : Type u_3} [MeasurableSpace X]
  [MeasurableSpace Ξ] [NormedAddCommGroup Y] [InnerProductSpace ℂ Y]
  (μ : MeasureTheory.Measure X) [MeasureTheory.SFinite μ]
  {ψ : X → Ξ → Y} {f : X → Y}
  (hψ : MeasureTheory.StronglyMeasurable (Function.uncurry ψ))
  (hf : MeasureTheory.AEStronglyMeasurable f μ) :
  MeasureTheory.StronglyMeasurable (LeanRidgelet.bochnerRidgelet μ ψ f)

theorem LeanRidgelet.stronglyMeasurable_bochnerRidgelet.{u_1,
    u_2, u_3}
  {X : Type u_1} {Ξ : Type u_2}
  {Y : Type u_3} [MeasurableSpace X]
  [MeasurableSpace Ξ]
  [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  (μ : MeasureTheory.Measure X)
  [MeasureTheory.SFinite μ]
  {ψ : X → Ξ → Y} {f : X → Y}
  (hψ :
    MeasureTheory.StronglyMeasurable
      (Function.uncurry ψ))
  (hf :
    MeasureTheory.AEStronglyMeasurable f
      μ) :
  MeasureTheory.StronglyMeasurable
    (LeanRidgelet.bochnerRidgelet μ ψ f)

**The analysis transform is measurable in the parameter, without a parameter measure.**  If the
feature is jointly strongly measurable -- not merely almost everywhere so for some product measure
-- then `LeanRidgelet.bochnerRidgelet` is strongly measurable on the parameter space, and only the
data measure is involved.  The data is still allowed to be measurable only almost everywhere,
because changing it on a null set of the data space does not change the integral for any parameter:
the two transforms are equal as functions, not just almost everywhere.

This is the form to use when the parameter measure is not yet fixed, for instance before a Tonelli
step that will replace the parameter measure by a product.

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.aestronglyMeasurable_bochnerSynthesis.{u_1, u_2, u_3}
  {X : Type u_1} {Ξ : Type u_2} {Y : Type u_3} [MeasurableSpace X]
  [MeasurableSpace Ξ] [NormedAddCommGroup Y] [InnerProductSpace ℂ Y]
  (μ : MeasureTheory.Measure Ξ) [MeasureTheory.SFinite μ]
  (ν : MeasureTheory.Measure X) [MeasureTheory.SFinite ν]
  {φ : X → Ξ → Y} {γ : Ξ → ℂ}
  (hφ :
    MeasureTheory.AEStronglyMeasurable (Function.uncurry φ) (ν.prod μ))
  (hγ : MeasureTheory.AEStronglyMeasurable γ μ) :
  MeasureTheory.AEStronglyMeasurable
    (LeanRidgelet.bochnerSynthesis μ φ γ) ν

theorem LeanRidgelet.aestronglyMeasurable_bochnerSynthesis.{u_1,
    u_2, u_3}
  {X : Type u_1} {Ξ : Type u_2}
  {Y : Type u_3} [MeasurableSpace X]
  [MeasurableSpace Ξ]
  [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  (μ : MeasureTheory.Measure Ξ)
  [MeasureTheory.SFinite μ]
  (ν : MeasureTheory.Measure X)
  [MeasureTheory.SFinite ν]
  {φ : X → Ξ → Y} {γ : Ξ → ℂ}
  (hφ :
    MeasureTheory.AEStronglyMeasurable
      (Function.uncurry φ) (ν.prod μ))
  (hγ :
    MeasureTheory.AEStronglyMeasurable γ
      μ) :
  MeasureTheory.AEStronglyMeasurable
    (LeanRidgelet.bochnerSynthesis μ φ γ)
    ν

**The synthesis is measurable in the data variable.**  The mirror of
`LeanRidgelet.aestronglyMeasurable_bochnerRidgelet`, and it needs nothing extra: the uncurried
feature `X × Ξ → Y` is already in the order the Fubini lemma wants, since the synthesis integrates
out the parameter and produces a function of the data, so no exchange of arguments occurs.  What
still costs σ-finiteness of the output measure is the coefficient, which has to be read as a
function on the product.

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.stronglyMeasurable_bochnerSynthesis.{u_1, u_2, u_3}
  {X : Type u_1} {Ξ : Type u_2} {Y : Type u_3} [MeasurableSpace X]
  [MeasurableSpace Ξ] [NormedAddCommGroup Y] [InnerProductSpace ℂ Y]
  (μ : MeasureTheory.Measure Ξ) [MeasureTheory.SFinite μ]
  {φ : X → Ξ → Y} {γ : Ξ → ℂ}
  (hφ : MeasureTheory.StronglyMeasurable (Function.uncurry φ))
  (hγ : MeasureTheory.AEStronglyMeasurable γ μ) :
  MeasureTheory.StronglyMeasurable (LeanRidgelet.bochnerSynthesis μ φ γ)

theorem LeanRidgelet.stronglyMeasurable_bochnerSynthesis.{u_1,
    u_2, u_3}
  {X : Type u_1} {Ξ : Type u_2}
  {Y : Type u_3} [MeasurableSpace X]
  [MeasurableSpace Ξ]
  [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  (μ : MeasureTheory.Measure Ξ)
  [MeasureTheory.SFinite μ]
  {φ : X → Ξ → Y} {γ : Ξ → ℂ}
  (hφ :
    MeasureTheory.StronglyMeasurable
      (Function.uncurry φ))
  (hγ :
    MeasureTheory.AEStronglyMeasurable γ
      μ) :
  MeasureTheory.StronglyMeasurable
    (LeanRidgelet.bochnerSynthesis μ φ γ)

**The synthesis is measurable in the data variable, without a data measure.**  The mirror of
`LeanRidgelet.stronglyMeasurable_bochnerRidgelet`: a jointly strongly measurable feature and an
almost everywhere measurable coefficient give a strongly measurable synthesis, with no measure on
the data space.

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.continuous_uncurry_quadraticArgument.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  Continuous (Function.uncurry LeanRidgelet.quadraticArgument)

theorem LeanRidgelet.continuous_uncurry_quadraticArgument.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  Continuous
    (Function.uncurry
      LeanRidgelet.quadraticArgument)

**The quadratic argument is jointly continuous.**  The scalar functional
`⟪x, A x⟫ + ⟪x, b⟫ + c` of `LeanRidgelet.quadraticArgument` is continuous in the pair of the data
and the parameter.  The three summands are continuous for the same reason in three degrees of
difficulty: the constant coefficient is a coordinate projection, the linear term is the inner
product of two projections, and the quadratic term is the inner product of a projection with the
evaluation `(A, x) ↦ A x`, which is continuous in the pair because it is a bounded bilinear map
(`isBoundedBilinearMap_apply`).  The symmetric coefficient enters that evaluation through the
coercion of `LeanRidgelet.QuadraticSymmetric E` into `E →L[ℝ] E`, which is continuous because the
subtype carries the induced topology.

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.continuous_uncurry_quadraticVectorFeature.{u_1, u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2} [NormedAddCommGroup Y]
  {σ : ℝ → Y} (hσ : Continuous σ) :
  Continuous (Function.uncurry (LeanRidgelet.quadraticVectorFeature σ))

theorem LeanRidgelet.continuous_uncurry_quadraticVectorFeature.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2}
  [NormedAddCommGroup Y] {σ : ℝ → Y}
  (hσ : Continuous σ) :
  Continuous
    (Function.uncurry
      (LeanRidgelet.quadraticVectorFeature
        σ))

**The quadratic feature is jointly continuous for a continuous activation.**  Immediate from
`LeanRidgelet.continuous_uncurry_quadraticArgument`, the feature being the activation composed with
the scalar argument.

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.measurable_uncurry_quadraticArgument.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] :
  Measurable (Function.uncurry LeanRidgelet.quadraticArgument)

theorem LeanRidgelet.measurable_uncurry_quadraticArgument.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric
        E)] :
  Measurable
    (Function.uncurry
      LeanRidgelet.quadraticArgument)

**The quadratic argument is jointly measurable.**  The Borel structures on the data space and on
the symmetric coefficients make the jointly continuous scalar argument jointly measurable.

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.stronglyMeasurable_uncurry_quadraticVectorFeature.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2} [NormedAddCommGroup Y]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] {σ : ℝ → Y}
  (hσ : MeasureTheory.StronglyMeasurable σ) :
  MeasureTheory.StronglyMeasurable
    (Function.uncurry (LeanRidgelet.quadraticVectorFeature σ))

theorem LeanRidgelet.stronglyMeasurable_uncurry_quadraticVectorFeature.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2}
  [NormedAddCommGroup Y]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {σ : ℝ → Y}
  (hσ :
    MeasureTheory.StronglyMeasurable σ) :
  MeasureTheory.StronglyMeasurable
    (Function.uncurry
      (LeanRidgelet.quadraticVectorFeature
        σ))

**The quadratic feature is jointly strongly measurable for a strongly measurable activation.**
This is the hypothesis that
`LeanRidgelet.stronglyMeasurable_bochnerRidgelet_quadraticVectorFeature` consumes; for a
scalar-valued activation it is the same as plain measurability, and a continuous activation
satisfies it through `LeanRidgelet.continuous_uncurry_quadraticVectorFeature`.

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.measurable_uncurry_quadraticVectorFeature.{u_1, u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2} [MeasurableSpace E]
  [BorelSpace E] [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] [MeasurableSpace Y]
  {σ : ℝ → Y} (hσ : Measurable σ) :
  Measurable (Function.uncurry (LeanRidgelet.quadraticVectorFeature σ))

theorem LeanRidgelet.measurable_uncurry_quadraticVectorFeature.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2}
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [MeasurableSpace Y] {σ : ℝ → Y}
  (hσ : Measurable σ) :
  Measurable
    (Function.uncurry
      (LeanRidgelet.quadraticVectorFeature
        σ))

**The quadratic feature is jointly measurable for a measurable activation.**  The plain
measurable form, for an activation into a bare measurable space.

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.stronglyMeasurable_bochnerRidgelet_quadraticVectorFeature.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2} [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (μ : MeasureTheory.Measure E) [MeasureTheory.SFinite μ] {σ : ℝ → Y}
  (hσ : MeasureTheory.StronglyMeasurable σ) {f : E → Y}
  (hf : MeasureTheory.AEStronglyMeasurable f μ) :
  MeasureTheory.StronglyMeasurable
    (LeanRidgelet.bochnerRidgelet μ
      (LeanRidgelet.quadraticVectorFeature σ) f)

theorem LeanRidgelet.stronglyMeasurable_bochnerRidgelet_quadraticVectorFeature.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2}
  [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (μ : MeasureTheory.Measure E)
  [MeasureTheory.SFinite μ] {σ : ℝ → Y}
  (hσ :
    MeasureTheory.StronglyMeasurable σ)
  {f : E → Y}
  (hf :
    MeasureTheory.AEStronglyMeasurable f
      μ) :
  MeasureTheory.StronglyMeasurable
    (LeanRidgelet.bochnerRidgelet μ
      (LeanRidgelet.quadraticVectorFeature
        σ)
      f)

**The analysis transform of the quadratic feature is measurable in the parameter.**  The
instantiation of `LeanRidgelet.stronglyMeasurable_bochnerRidgelet` at the quadratic feature: for a
strongly measurable activation and measurable data, the analysis transform is a strongly measurable
function of the quadratic parameter `(A, b, c)`.  No parameter measure enters.

theoremdefined in LeanRidgelet/HA/BochnerMeasurability.lean

complete

theorem LeanRidgelet.aestronglyMeasurable_bochnerRidgelet_quadraticVectorFeature.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2} [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (μ : MeasureTheory.Measure E) [MeasureTheory.SFinite μ]
  (ν : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  {σ : ℝ → Y} (hσ : MeasureTheory.StronglyMeasurable σ) {f : E → Y}
  (hf : MeasureTheory.AEStronglyMeasurable f μ) :
  MeasureTheory.AEStronglyMeasurable
    (LeanRidgelet.bochnerRidgelet μ
      (LeanRidgelet.quadraticVectorFeature σ) f)
    ν

theorem LeanRidgelet.aestronglyMeasurable_bochnerRidgelet_quadraticVectorFeature.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {Y : Type u_2}
  [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (μ : MeasureTheory.Measure E)
  [MeasureTheory.SFinite μ]
  (ν :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  {σ : ℝ → Y}
  (hσ :
    MeasureTheory.StronglyMeasurable σ)
  {f : E → Y}
  (hf :
    MeasureTheory.AEStronglyMeasurable f
      μ) :
  MeasureTheory.AEStronglyMeasurable
    (LeanRidgelet.bochnerRidgelet μ
      (LeanRidgelet.quadraticVectorFeature
        σ)
      f)
    ν

**The analysis transform of the quadratic feature is measurable in the parameter, almost
everywhere form.**  The same statement for an arbitrary measure on the quadratic parameter space,
which is the shape a Tonelli argument over the parameters consumes.  Since the previous theorem
gives genuine strong measurability, this holds for every parameter measure with no σ-finiteness
hypothesis on it -- in particular for `LeanRidgelet.quadraticRelativeMeasure`.

The transforms are raw Bochner integrals with no hypotheses attached, so nothing about them is measurable until said. The general statements are the ones the parametric-integral lemma of Mathlib gives, with the feature entered in the uncurried form that lemma consumes, and they are stated in the abstract setting so that the ridge feature is covered too. For the quadratic feature the joint hypothesis is not an assumption: its scalar argument is jointly continuous, because evaluating a continuous linear map is continuous in the operator and the vector together, so composing with a continuous or measurable activation gives what is needed. This is what the integration of the slice-wise weighted estimate was missing.

Lemma4.5.9

Statement uses 2

Statement dependency previews

Preview

Theorem 4.5.4

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

Used by 3

Reverse dependency previews

Preview

Lemma 4.5.10

Loading preview

Reverse dependency preview content is loaded from the rendered-fragment cache.

✓L∃∀N

The parameter measure factors over the constant coefficient. Under the associativity transport it is a positive multiple of a product of the corresponding measure on the other two coefficients with Lebesgue measure, so an integral against it is an iterated integral.

Lean code for Lemma4.5.9●13 declarations

defdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

def LeanRidgelet.quadraticBaseDet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (p : LeanRidgelet.QuadraticSymmetric E × E) : ℝ

def LeanRidgelet.quadraticBaseDet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  ℝ

Implementation after :=

:= (p.1 : E →L[ℝ] E).det

The determinant of the symmetric coefficient of a *base parameter* `(A, b)`, that is, of a
quadratic parameter with the constant coefficient dropped.  This is
`LeanRidgelet.quadraticSymmetricDet` read on the first two coordinates only.

theoremdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

theorem LeanRidgelet.quadraticSymmetricDet_eq_quadraticBaseDet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticSymmetricDet ξ =
    LeanRidgelet.quadraticBaseDet (ξ.1, ξ.2.1)

theorem LeanRidgelet.quadraticSymmetricDet_eq_quadraticBaseDet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticSymmetricDet ξ =
    LeanRidgelet.quadraticBaseDet
      (ξ.1, ξ.2.1)

The symmetric determinant of a quadratic parameter reads only its base parameter.  This is the
factorization that makes the whole file work.

defdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

def LeanRidgelet.quadraticBaseNondegenerate.{u_2} (E : Type u_2)
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] : Set (LeanRidgelet.QuadraticSymmetric E × E)

def LeanRidgelet.quadraticBaseNondegenerate.{u_2}
  (E : Type u_2) [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  Set
    (LeanRidgelet.QuadraticSymmetric E ×
      E)

Implementation after :=

:=
  {p | quadraticBaseDet p ≠ 0}

The nondegenerate locus of base parameters.  Compare
`LeanRidgelet.quadraticNondegenerate`.

theoremdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

theorem LeanRidgelet.quadraticNondegenerate_eq_preimage.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  LeanRidgelet.quadraticNondegenerate E =
    {ξ | (ξ.1, ξ.2.1) ∈ LeanRidgelet.quadraticBaseNondegenerate E}

theorem LeanRidgelet.quadraticNondegenerate_eq_preimage.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  LeanRidgelet.quadraticNondegenerate E =
    {ξ |
      (ξ.1, ξ.2.1) ∈
        LeanRidgelet.quadraticBaseNondegenerate
          E}

The nondegenerate locus of quadratic parameters is the cylinder over the nondegenerate locus of
base parameters.

defdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

def LeanRidgelet.quadraticBaseWeight.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (p : LeanRidgelet.QuadraticSymmetric E × E) : ENNReal

def LeanRidgelet.quadraticBaseWeight.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  ENNReal

Implementation after :=

:=
  ENNReal.ofReal (|quadraticBaseDet p| ^ (-(Module.finrank ℝ E + 1 : ℝ) / 2))

The relatively invariant weight on the base parameter space, with the same exponent
`-(m + 1) / 2` as `LeanRidgelet.quadraticRelativeWeight`.

theoremdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

theorem LeanRidgelet.quadraticRelativeWeight_eq_quadraticBaseWeight.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  LeanRidgelet.quadraticRelativeWeight = fun ξ ↦
    LeanRidgelet.quadraticBaseWeight (ξ.1, ξ.2.1)

theorem LeanRidgelet.quadraticRelativeWeight_eq_quadraticBaseWeight.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  LeanRidgelet.quadraticRelativeWeight =
    fun ξ ↦
    LeanRidgelet.quadraticBaseWeight
      (ξ.1, ξ.2.1)

The relatively invariant weight of a quadratic parameter reads only its base parameter.

defdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

def LeanRidgelet.quadraticBaseRelativeMeasure.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (κ : MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E)) :
  MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E)

def LeanRidgelet.quadraticBaseRelativeMeasure.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (κ :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticSymmetric E ×
        E)) :
  MeasureTheory.Measure
    (LeanRidgelet.QuadraticSymmetric E ×
      E)

Implementation after :=

:=
  (κ.restrict (quadraticBaseNondegenerate E)).withDensity quadraticBaseWeight

**The base parameter measure.**  An additive Haar measure `κ` on the base parameter space,
restricted to the nondegenerate locus and weighted by `LeanRidgelet.quadraticBaseWeight`; the exact
analogue of `LeanRidgelet.quadraticRelativeMeasure` one factor down.

theoremdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

theorem LeanRidgelet.measurable_quadraticBaseDet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] :
  Measurable LeanRidgelet.quadraticBaseDet

theorem LeanRidgelet.measurable_quadraticBaseDet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric
        E)] :
  Measurable LeanRidgelet.quadraticBaseDet

The base symmetric determinant is measurable, being continuous in the symmetric
coefficient.

theoremdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

theorem LeanRidgelet.measurableSet_quadraticBaseNondegenerate.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] :
  MeasurableSet (LeanRidgelet.quadraticBaseNondegenerate E)

theorem LeanRidgelet.measurableSet_quadraticBaseNondegenerate.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric
        E)] :
  MeasurableSet
    (LeanRidgelet.quadraticBaseNondegenerate
      E)

The nondegenerate locus of base parameters is measurable.

theoremdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

theorem LeanRidgelet.measurable_quadraticBaseWeight.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] :
  Measurable LeanRidgelet.quadraticBaseWeight

theorem LeanRidgelet.measurable_quadraticBaseWeight.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric
        E)] :
  Measurable
    LeanRidgelet.quadraticBaseWeight

The base weight is measurable.

theoremdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

theorem LeanRidgelet.exists_map_prodAssoc_symm_quadraticRelativeMeasure_eq_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (κ : MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E))
  [κ.IsAddHaarMeasure] :
  ∃ c,
    c ≠ 0 ∧
      c ≠ ⊤ ∧
        MeasureTheory.Measure.map (⇑MeasurableEquiv.prodAssoc.symm)
            (LeanRidgelet.quadraticRelativeMeasure lam) =
          c •
            (LeanRidgelet.quadraticBaseRelativeMeasure κ).prod
              MeasureTheory.volume

theorem LeanRidgelet.exists_map_prodAssoc_symm_quadraticRelativeMeasure_eq_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (κ :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticSymmetric E ×
        E))
  [κ.IsAddHaarMeasure] :
  ∃ c,
    c ≠ 0 ∧
      c ≠ ⊤ ∧
        MeasureTheory.Measure.map
            (⇑MeasurableEquiv.prodAssoc.symm)
            (LeanRidgelet.quadraticRelativeMeasure
              lam) =
          c •
            (LeanRidgelet.quadraticBaseRelativeMeasure
                  κ).prod
              MeasureTheory.volume

**The quadratic parameter measure factors over the constant coefficient.**  Transported along
the associativity equivalence, `LeanRidgelet.quadraticRelativeMeasure lam` is a positive finite
multiple of the product of the base parameter measure of any additive Haar measure `κ` on
`QuadraticSymmetric E × E` with the Lebesgue measure of the constant coefficient.  The constant is
not determined: `lam` and `κ` are each only fixed up to a scalar.

theoremdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

theorem LeanRidgelet.lintegral_quadraticRelativeMeasure_of_map_eq_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  {lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)}
  {κ : MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E)}
  {c : ENNReal}
  (hmap :
    MeasureTheory.Measure.map (⇑MeasurableEquiv.prodAssoc.symm)
        (LeanRidgelet.quadraticRelativeMeasure lam) =
      c •
        (LeanRidgelet.quadraticBaseRelativeMeasure κ).prod
          MeasureTheory.volume)
  {g : LeanRidgelet.QuadraticParameter E → ENNReal}
  (hg : Measurable g) :
  ∫⁻ (ξ : LeanRidgelet.QuadraticParameter E),
      g ξ ∂LeanRidgelet.quadraticRelativeMeasure lam =
    c *
      ∫⁻ (p : LeanRidgelet.QuadraticSymmetric E × E),
        ∫⁻ (t : ℝ),
          g (p.1, p.2, t) ∂LeanRidgelet.quadraticBaseRelativeMeasure κ

theorem LeanRidgelet.lintegral_quadraticRelativeMeasure_of_map_eq_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E)}
  {κ :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticSymmetric E ×
        E)}
  {c : ENNReal}
  (hmap :
    MeasureTheory.Measure.map
        (⇑MeasurableEquiv.prodAssoc.symm)
        (LeanRidgelet.quadraticRelativeMeasure
          lam) =
      c •
        (LeanRidgelet.quadraticBaseRelativeMeasure
              κ).prod
          MeasureTheory.volume)
  {g :
    LeanRidgelet.QuadraticParameter E →
      ENNReal}
  (hg : Measurable g) :
  ∫⁻ (ξ :
      LeanRidgelet.QuadraticParameter E),
      g
        ξ ∂LeanRidgelet.quadraticRelativeMeasure
        lam =
    c *
      ∫⁻ (p :
        LeanRidgelet.QuadraticSymmetric
            E ×
          E),
        ∫⁻ (t : ℝ),
          g
            (p.1, p.2,
              t) ∂LeanRidgelet.quadraticBaseRelativeMeasure
          κ

**The factorization spent on an integral.**  Given the factorization, a lower Lebesgue integral
against the quadratic parameter measure is the constant times the iterated integral over the base
parameter and the constant coefficient — the form a Tonelli argument consumes.  Only the outer
measure is abstract; the inner one is the Lebesgue measure of the constant coefficient.

theoremdefined in LeanRidgelet/HA/QuadraticParameterFactor.lean

complete

theorem LeanRidgelet.exists_lintegral_quadraticRelativeMeasure_eq_mul_lintegral.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (κ : MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E))
  [κ.IsAddHaarMeasure] {g : LeanRidgelet.QuadraticParameter E → ENNReal}
  (hg : Measurable g) :
  ∃ c,
    c ≠ 0 ∧
      c ≠ ⊤ ∧
        ∫⁻ (ξ : LeanRidgelet.QuadraticParameter E),
            g ξ ∂LeanRidgelet.quadraticRelativeMeasure lam =
          c *
            ∫⁻ (p : LeanRidgelet.QuadraticSymmetric E × E),
              ∫⁻ (t : ℝ),
                g
                  (p.1, p.2,
                    t) ∂LeanRidgelet.quadraticBaseRelativeMeasure κ

theorem LeanRidgelet.exists_lintegral_quadraticRelativeMeasure_eq_mul_lintegral.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (κ :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticSymmetric E ×
        E))
  [κ.IsAddHaarMeasure]
  {g :
    LeanRidgelet.QuadraticParameter E →
      ENNReal}
  (hg : Measurable g) :
  ∃ c,
    c ≠ 0 ∧
      c ≠ ⊤ ∧
        ∫⁻ (ξ :
            LeanRidgelet.QuadraticParameter
              E),
            g
              ξ ∂LeanRidgelet.quadraticRelativeMeasure
              lam =
          c *
            ∫⁻ (p :
              LeanRidgelet.QuadraticSymmetric
                  E ×
                E),
              ∫⁻ (t : ℝ),
                g
                  (p.1, p.2,
                    t) ∂LeanRidgelet.quadraticBaseRelativeMeasure
                κ

The two previous results combined: the integral against the quadratic parameter measure is a
positive finite multiple of the iterated integral over the base parameter and the constant
coefficient, for any additive Haar measures `lam` and `κ`.

The premise is that the degenerate locus and the weight read only the symmetric coefficient, which they do — both are conditions on its determinant, and the three compatibility statements are definitional. The general reassociation of the Mathlib candidate layer then applies, with one hypothesis the caller supplies: an additive Haar measure on the first two coefficients to compare against. There is no canonical choice, the parameter measure itself being abstract, and changing it only changes the constant.

This removes the measure obstruction to integrating the slice-wise weighted estimate. It does not integrate it: that is recorded above.

Lemma4.5.10

Statement uses 2

Statement dependency previews

Preview

Lemma 4.5.7

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

Used by 4

Reverse dependency previews

Preview

Lemma 4.5.11

Loading preview

Reverse dependency preview content is loaded from the rendered-fragment cache.

✓L∃∀N

The parameter action is a shear in the constant coefficient. The first two coefficients transform without reference to the third, and the third is itself plus a shift depending only on the other two — so differentiation in it commutes with the action, with no factor, and the Sobolev seminorms in it are carried along.

Lean code for Lemma4.5.10●19 declarations

defdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

def LeanRidgelet.quadraticBaseLinearEquiv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  (LeanRidgelet.QuadraticSymmetric E × E) ≃ₗ[ℝ]
    LeanRidgelet.QuadraticSymmetric E × E

def LeanRidgelet.quadraticBaseLinearEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  (LeanRidgelet.QuadraticSymmetric E ×
      E) ≃ₗ[ℝ]
    LeanRidgelet.QuadraticSymmetric E × E

Implementation after :=

:=
  (quadraticCongr g.linear.symm).skewProd g.linear.symm.adjoint
    ((-2 : ℝ) • quadraticShearVector g)

**The action on the base parameter.**  The first two coefficients of `g • (A, b, c)`, as a
linear automorphism of the base parameter `(A, b)`: congruence by the inverse linear part on the
symmetric coefficient, and the adjoint of the inverse linear part on the linear coefficient, skewed
by the translation block `LeanRidgelet.quadraticShearVector`.  The constant coefficient does not
enter, which is the first half of the shear.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.quadraticBaseLinearEquiv_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (A : LeanRidgelet.QuadraticSymmetric E) (b : E) :
  (LeanRidgelet.quadraticBaseLinearEquiv g) (A, b) =
    ((LeanRidgelet.quadraticCongr g.linear.symm) A,
      g.linear.symm.adjoint b +
        -2 • (LeanRidgelet.quadraticShearVector g) A)

theorem LeanRidgelet.quadraticBaseLinearEquiv_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (A : LeanRidgelet.QuadraticSymmetric E)
  (b : E) :
  (LeanRidgelet.quadraticBaseLinearEquiv
        g)
      (A, b) =
    ((LeanRidgelet.quadraticCongr
          g.linear.symm)
        A,
      g.linear.symm.adjoint b +
        -2 •
          (LeanRidgelet.quadraticShearVector
              g)
            A)

The two components of the base action, read off the skew product.

defdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

def LeanRidgelet.quadraticConstShift.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : LeanRidgelet.QuadraticSymmetric E × E →ₗ[ℝ] ℝ

def LeanRidgelet.quadraticConstShift.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.QuadraticSymmetric E ×
      E →ₗ[ℝ]
    ℝ

Implementation after :=

:=
  (quadraticLinearShear g).comp (LinearMap.snd ℝ (QuadraticSymmetric E) E) +
    ((innerSL ℝ (g 0)).toLinearMap.comp (quadraticShearVector g)).comp
      (LinearMap.fst ℝ (QuadraticSymmetric E) E)

**The shift of the constant coefficient.**  The amount by which the parameter action of `g`
displaces the constant coefficient.  It is a linear functional of the base parameter alone, which is
the second half of the shear.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.quadraticConstShift_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (A : LeanRidgelet.QuadraticSymmetric E) (b : E) :
  (LeanRidgelet.quadraticConstShift g) (A, b) =
    (LeanRidgelet.quadraticLinearShear g) b +
      inner ℝ (g 0) ((LeanRidgelet.quadraticShearVector g) A)

theorem LeanRidgelet.quadraticConstShift_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (A : LeanRidgelet.QuadraticSymmetric E)
  (b : E) :
  (LeanRidgelet.quadraticConstShift g)
      (A, b) =
    (LeanRidgelet.quadraticLinearShear g)
        b +
      inner ℝ (g 0)
        ((LeanRidgelet.quadraticShearVector
            g)
          A)

The shift of the constant coefficient, as the sum of its linear and symmetric blocks.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.quadraticParameterLinearEquiv_apply_shear.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (c : ℝ) :
  (LeanRidgelet.quadraticParameterLinearEquiv g) (A, b, c) =
    (((LeanRidgelet.quadraticBaseLinearEquiv g) (A, b)).1,
      ((LeanRidgelet.quadraticBaseLinearEquiv g) (A, b)).2,
      c + (LeanRidgelet.quadraticConstShift g) (A, b))

theorem LeanRidgelet.quadraticParameterLinearEquiv_apply_shear.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (A : LeanRidgelet.QuadraticSymmetric E)
  (b : E) (c : ℝ) :
  (LeanRidgelet.quadraticParameterLinearEquiv
        g)
      (A, b, c) =
    (((LeanRidgelet.quadraticBaseLinearEquiv
            g)
          (A, b)).1,
      ((LeanRidgelet.quadraticBaseLinearEquiv
            g)
          (A, b)).2,
      c +
        (LeanRidgelet.quadraticConstShift
            g)
          (A, b))

**The parameter action is a shear in the constant coefficient.**  For every affine `g` the
parameter action sends `(A, b, c)` to `(A' (A, b), b' (A, b), c + s (A, b))`: the first two
components are a function of the first two components alone, and the third is the old constant
coefficient plus a quantity independent of it.  The shear is definitional: against
`LeanRidgelet.quadraticParameterLinearEquiv_apply` the first two components hold by `rfl` and the
third needs exactly one reassociation of addition, as the proof shows.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.quadraticParameterLinearEquiv_eq_shear.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (ξ : LeanRidgelet.QuadraticParameter E) :
  (LeanRidgelet.quadraticParameterLinearEquiv g) ξ =
    (((LeanRidgelet.quadraticBaseLinearEquiv g) (ξ.1, ξ.2.1)).1,
      ((LeanRidgelet.quadraticBaseLinearEquiv g) (ξ.1, ξ.2.1)).2,
      ξ.2.2 + (LeanRidgelet.quadraticConstShift g) (ξ.1, ξ.2.1))

theorem LeanRidgelet.quadraticParameterLinearEquiv_eq_shear.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  (LeanRidgelet.quadraticParameterLinearEquiv
        g)
      ξ =
    (((LeanRidgelet.quadraticBaseLinearEquiv
            g)
          (ξ.1, ξ.2.1)).1,
      ((LeanRidgelet.quadraticBaseLinearEquiv
            g)
          (ξ.1, ξ.2.1)).2,
      ξ.2.2 +
        (LeanRidgelet.quadraticConstShift
            g)
          (ξ.1, ξ.2.1))

The shear form of the parameter action at an unsplit parameter.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.exists_quadraticParameterLinearEquiv_shear.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E) :
  ∃ F s,
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (c : ℝ),
      (LeanRidgelet.quadraticParameterLinearEquiv g) (A, b, c) =
        ((F (A, b)).1, (F (A, b)).2, c + s (A, b))

theorem LeanRidgelet.exists_quadraticParameterLinearEquiv_shear.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  ∃ F s,
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (c : ℝ),
      (LeanRidgelet.quadraticParameterLinearEquiv
            g)
          (A, b, c) =
        ((F (A, b)).1, (F (A, b)).2,
          c + s (A, b))

**The shear, in existential form.**  For every affine `g` there is a linear automorphism `F` of
the base parameter and a linear functional `s` on it such that the parameter action of `g` is
`(A, b, c) ↦ ((F (A, b)).1, (F (A, b)).2, c + s (A, b))`, uniformly in `c`.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.quadraticParameterLinearEquiv_apply_const_add.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (c d : ℝ) :
  (LeanRidgelet.quadraticParameterLinearEquiv g) (A, b, c + d) =
    (((LeanRidgelet.quadraticParameterLinearEquiv g) (A, b, c)).1,
      ((LeanRidgelet.quadraticParameterLinearEquiv g) (A, b, c)).2.1,
      ((LeanRidgelet.quadraticParameterLinearEquiv g) (A, b, c)).2.2 +
        d)

theorem LeanRidgelet.quadraticParameterLinearEquiv_apply_const_add.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (A : LeanRidgelet.QuadraticSymmetric E)
  (b : E) (c d : ℝ) :
  (LeanRidgelet.quadraticParameterLinearEquiv
        g)
      (A, b, c + d) =
    (((LeanRidgelet.quadraticParameterLinearEquiv
            g)
          (A, b, c)).1,
      ((LeanRidgelet.quadraticParameterLinearEquiv
              g)
            (A, b, c)).2.1,
      ((LeanRidgelet.quadraticParameterLinearEquiv
                g)
              (A, b, c)).2.2 +
        d)

**The shear, in difference form.**  Changing the constant coefficient of the argument leaves the
first two components of the transformed parameter alone and moves its constant coefficient by
exactly the same amount.  No reference to the explicit blocks is made.

defdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

def LeanRidgelet.quadraticConstSlice.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (T : LeanRidgelet.QuadraticParameter E → ℂ)
  (p : LeanRidgelet.QuadraticSymmetric E × E) : ℝ → ℂ

def LeanRidgelet.quadraticConstSlice.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  ℝ → ℂ

Implementation after :=

:=
  fun t ↦ T (p.1, p.2, t)

The slice of a function on the quadratic parameter space in the constant coefficient, at a
frozen base parameter.  This is the object `LeanRidgelet.HA.QuadraticWeighted` estimates.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.quadraticConstSlice_comp_smul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (T : LeanRidgelet.QuadraticParameter E → ℂ) (g : E ≃ᵃ[ℝ] E)
  (p : LeanRidgelet.QuadraticSymmetric E × E) :
  LeanRidgelet.quadraticConstSlice (fun η ↦ T (g • η)) p = fun t ↦
    LeanRidgelet.quadraticConstSlice T
      ((LeanRidgelet.quadraticBaseLinearEquiv g) p)
      (t + (LeanRidgelet.quadraticConstShift g) p)

theorem LeanRidgelet.quadraticConstSlice_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (g : E ≃ᵃ[ℝ] E)
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  LeanRidgelet.quadraticConstSlice
      (fun η ↦ T (g • η)) p =
    fun t ↦
    LeanRidgelet.quadraticConstSlice T
      ((LeanRidgelet.quadraticBaseLinearEquiv
          g)
        p)
      (t +
        (LeanRidgelet.quadraticConstShift
            g)
          p)

**The pull-back is a translated slice.**  Pulling back along the parameter action and then
slicing in the constant coefficient gives the slice at the transformed base parameter, translated by
the shift.  Nothing else happens: in particular the last coordinate is not rescaled.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.hasDerivAt_quadraticConstSlice_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {T : LeanRidgelet.QuadraticParameter E → ℂ}
  {T' : ℂ} (g : E ≃ᵃ[ℝ] E) (p : LeanRidgelet.QuadraticSymmetric E × E)
  (c : ℝ)
  (h :
    HasDerivAt
      (LeanRidgelet.quadraticConstSlice T
        ((LeanRidgelet.quadraticBaseLinearEquiv g) p))
      T' (c + (LeanRidgelet.quadraticConstShift g) p)) :
  HasDerivAt (LeanRidgelet.quadraticConstSlice (fun η ↦ T (g • η)) p) T'
    c

theorem LeanRidgelet.hasDerivAt_quadraticConstSlice_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  {T' : ℂ} (g : E ≃ᵃ[ℝ] E)
  (p :
    LeanRidgelet.QuadraticSymmetric E × E)
  (c : ℝ)
  (h :
    HasDerivAt
      (LeanRidgelet.quadraticConstSlice T
        ((LeanRidgelet.quadraticBaseLinearEquiv
            g)
          p))
      T'
      (c +
        (LeanRidgelet.quadraticConstShift
            g)
          p)) :
  HasDerivAt
    (LeanRidgelet.quadraticConstSlice
      (fun η ↦ T (g • η)) p)
    T' c

**Commutation with differentiation, at a point.**  If the slice of `T` at the transformed base
parameter has derivative `T'` at the shifted point, then the slice of the pull-back of `T` has the
*same* derivative `T'` at the unshifted point.  No factor appears, because the shear translates the
constant coefficient instead of rescaling it.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.iteratedDeriv_quadraticConstSlice_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T : LeanRidgelet.QuadraticParameter E → ℂ) (g : E ≃ᵃ[ℝ] E)
  (p : LeanRidgelet.QuadraticSymmetric E × E) :
  iteratedDeriv j
      (LeanRidgelet.quadraticConstSlice (fun η ↦ T (g • η)) p) =
    fun c ↦
    iteratedDeriv j
      (LeanRidgelet.quadraticConstSlice T
        ((LeanRidgelet.quadraticBaseLinearEquiv g) p))
      (c + (LeanRidgelet.quadraticConstShift g) p)

theorem LeanRidgelet.iteratedDeriv_quadraticConstSlice_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (g : E ≃ᵃ[ℝ] E)
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  iteratedDeriv j
      (LeanRidgelet.quadraticConstSlice
        (fun η ↦ T (g • η)) p) =
    fun c ↦
    iteratedDeriv j
      (LeanRidgelet.quadraticConstSlice T
        ((LeanRidgelet.quadraticBaseLinearEquiv
            g)
          p))
      (c +
        (LeanRidgelet.quadraticConstShift
            g)
          p)

**Iterating the commutation.**  The `j`-th derivative in the constant coefficient of the slice
of the pull-back is the `j`-th derivative of the slice at the transformed base parameter, evaluated
at the shifted point.  Compare
`LeanRidgelet.iteratedDeriv_bochnerRidgelet_quadraticVectorFeature`, which is stated the same way,
as an equality of functions of the constant coefficient.  Unlike that result this one needs no
hypotheses: translation invariance of `iteratedDeriv` holds for every function.

defdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

def LeanRidgelet.quadraticConstIteratedDeriv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (j : ℕ) (T : LeanRidgelet.QuadraticParameter E → ℂ) :
  LeanRidgelet.QuadraticParameter E → ℂ

def LeanRidgelet.quadraticConstIteratedDeriv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  LeanRidgelet.QuadraticParameter E → ℂ

Implementation after :=

:=
  fun ξ ↦ iteratedDeriv j (quadraticConstSlice T (ξ.1, ξ.2.1)) ξ.2.2

The `j`-th derivative in the constant coefficient of a function on the quadratic parameter
space, again as a function on the parameter space.  This is the object whose `L²` norm is the `j`-th
`Γ`-seminorm.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.quadraticConstIteratedDeriv_comp_smul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (j : ℕ) (T : LeanRidgelet.QuadraticParameter E → ℂ) (g : E ≃ᵃ[ℝ] E)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticConstIteratedDeriv j (fun η ↦ T (g • η)) ξ =
    LeanRidgelet.quadraticConstIteratedDeriv j T (g • ξ)

theorem LeanRidgelet.quadraticConstIteratedDeriv_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (g : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticConstIteratedDeriv
      j (fun η ↦ T (g • η)) ξ =
    LeanRidgelet.quadraticConstIteratedDeriv
      j T (g • ξ)

**Equivariance of the derivative in the constant coefficient.**  The `j`-th derivative in the
constant coefficient of a pull-back is the pull-back of the `j`-th derivative, with no factor and at
every parameter.  This is `LeanRidgelet.iteratedDeriv_quadraticConstSlice_comp_smul` read on the
whole parameter space, and it is what makes the `Γ`-seminorms below transform like the parameter
measure.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.quadraticConstIteratedDeriv_comp_smul_eq.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T : LeanRidgelet.QuadraticParameter E → ℂ) (g : E ≃ᵃ[ℝ] E) :
  (LeanRidgelet.quadraticConstIteratedDeriv j fun η ↦ T (g • η)) =
    fun ξ ↦ LeanRidgelet.quadraticConstIteratedDeriv j T (g • ξ)

theorem LeanRidgelet.quadraticConstIteratedDeriv_comp_smul_eq.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (g : E ≃ᵃ[ℝ] E) :
  (LeanRidgelet.quadraticConstIteratedDeriv
      j fun η ↦ T (g • η)) =
    fun ξ ↦
    LeanRidgelet.quadraticConstIteratedDeriv
      j T (g • ξ)

The pull-back of the derivative in the constant coefficient, as a composition.  This is the form
the measure-theoretic statements consume.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.eLpNorm_iteratedDeriv_quadraticConstSlice_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T : LeanRidgelet.QuadraticParameter E → ℂ) (g : E ≃ᵃ[ℝ] E)
  (p : LeanRidgelet.QuadraticSymmetric E × E) (r : ENNReal)
  (hmeas :
    MeasureTheory.AEStronglyMeasurable
      (iteratedDeriv j
        (LeanRidgelet.quadraticConstSlice T
          ((LeanRidgelet.quadraticBaseLinearEquiv g) p)))
      MeasureTheory.volume) :
  MeasureTheory.eLpNorm
      (iteratedDeriv j
        (LeanRidgelet.quadraticConstSlice (fun η ↦ T (g • η)) p))
      r MeasureTheory.volume =
    MeasureTheory.eLpNorm
      (iteratedDeriv j
        (LeanRidgelet.quadraticConstSlice T
          ((LeanRidgelet.quadraticBaseLinearEquiv g) p)))
      r MeasureTheory.volume

theorem LeanRidgelet.eLpNorm_iteratedDeriv_quadraticConstSlice_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (g : E ≃ᵃ[ℝ] E)
  (p :
    LeanRidgelet.QuadraticSymmetric E × E)
  (r : ENNReal)
  (hmeas :
    MeasureTheory.AEStronglyMeasurable
      (iteratedDeriv j
        (LeanRidgelet.quadraticConstSlice
          T
          ((LeanRidgelet.quadraticBaseLinearEquiv
              g)
            p)))
      MeasureTheory.volume) :
  MeasureTheory.eLpNorm
      (iteratedDeriv j
        (LeanRidgelet.quadraticConstSlice
          (fun η ↦ T (g • η)) p))
      r MeasureTheory.volume =
    MeasureTheory.eLpNorm
      (iteratedDeriv j
        (LeanRidgelet.quadraticConstSlice
          T
          ((LeanRidgelet.quadraticBaseLinearEquiv
              g)
            p)))
      r MeasureTheory.volume

**Slice-wise invariance of the Sobolev seminorm.**  For a frozen base parameter, the `Lᵖ` norm
in the constant coefficient of the `j`-th derivative of the slice of a pull-back equals the `Lᵖ`
norm of the `j`-th derivative of the slice at the transformed base parameter, with constant exactly
`1`.  Only two inputs are used: the shear translates the constant coefficient, and Lebesgue measure
on the line is translation invariant.

This is the honest slice-wise form.  The constant `1` is not a normalization choice: the shift
enters only as a translation, and the modulus of the group appears not here but in the base
direction, where it is supplied by the quasi-invariance of the parameter measure; see
`LeanRidgelet.lintegral_enorm_quadraticConstIteratedDeriv_comp_smul`.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.lintegral_comp_smul_quadraticRelativeMeasure.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  {F : LeanRidgelet.QuadraticParameter E → ENNReal}
  (hF : Measurable F) :
  ∫⁻ (ξ : LeanRidgelet.QuadraticParameter E),
      F (g • ξ) ∂LeanRidgelet.quadraticRelativeMeasure lam =
    ↑‖LinearMap.det ↑g.linear‖₊ *
      ∫⁻ (ξ : LeanRidgelet.QuadraticParameter E),
        F ξ ∂LeanRidgelet.quadraticRelativeMeasure lam

theorem LeanRidgelet.lintegral_comp_smul_quadraticRelativeMeasure.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  {F :
    LeanRidgelet.QuadraticParameter E →
      ENNReal}
  (hF : Measurable F) :
  ∫⁻ (ξ :
      LeanRidgelet.QuadraticParameter E),
      F
        (g •
          ξ) ∂LeanRidgelet.quadraticRelativeMeasure
        lam =
    ↑‖LinearMap.det ↑g.linear‖₊ *
      ∫⁻ (ξ :
        LeanRidgelet.QuadraticParameter
          E),
        F
          ξ ∂LeanRidgelet.quadraticRelativeMeasure
          lam

**The parameter measure spent on a pull-back.**  A lower Lebesgue integral against the
relatively invariant parameter measure of a function pulled back along the parameter action of `g`
is `‖det L‖₊` times the integral of the function.  This is
`LeanRidgelet.quadraticRelativeParameter_group_map_eq_withDensity` at `g⁻¹`, whose constant density
inverts to `‖det L‖₊`.  Nothing about the shear is used here; it is the measure's own
quasi-invariance.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.lintegral_enorm_quadraticConstIteratedDeriv_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) (j : ℕ)
  {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : Measurable (LeanRidgelet.quadraticConstIteratedDeriv j T)) :
  ∫⁻ (ξ : LeanRidgelet.QuadraticParameter E),
      ‖LeanRidgelet.quadraticConstIteratedDeriv j (fun η ↦ T (g • η))
            ξ‖ₑ ^
        2 ∂LeanRidgelet.quadraticRelativeMeasure lam =
    ↑‖LinearMap.det ↑g.linear‖₊ *
      ∫⁻ (ξ : LeanRidgelet.QuadraticParameter E),
        ‖LeanRidgelet.quadraticConstIteratedDeriv j T ξ‖ₑ ^
          2 ∂LeanRidgelet.quadraticRelativeMeasure lam

theorem LeanRidgelet.lintegral_enorm_quadraticConstIteratedDeriv_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  (j : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT :
    Measurable
      (LeanRidgelet.quadraticConstIteratedDeriv
        j T)) :
  ∫⁻ (ξ :
      LeanRidgelet.QuadraticParameter E),
      ‖LeanRidgelet.quadraticConstIteratedDeriv
            j (fun η ↦ T (g • η)) ξ‖ₑ ^
        2 ∂LeanRidgelet.quadraticRelativeMeasure
        lam =
    ↑‖LinearMap.det ↑g.linear‖₊ *
      ∫⁻ (ξ :
        LeanRidgelet.QuadraticParameter
          E),
        ‖LeanRidgelet.quadraticConstIteratedDeriv
              j T ξ‖ₑ ^
          2 ∂LeanRidgelet.quadraticRelativeMeasure
          lam

**Global invariance of the Sobolev seminorm, up to the quasi-invariance constant.**  The `L²`
norm in the constant coefficient of the `j`-th derivative, integrated over the whole parameter space
against the relatively invariant parameter measure, is multiplied by exactly `‖det L‖₊` when the
argument is pulled back along the parameter action of `g`.  That constant is the one the parameter
measure contributes on its own, so these seminorms become invariant under exactly the same
square-root normalization of the pull-back that makes
`LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation` unitary.

The measurability hypothesis is on the derivative in the constant coefficient as a function of the
parameter; it is not available anywhere in this development and is carried as a hypothesis.

theoremdefined in LeanRidgelet/HA/QuadraticShear.lean

complete

theorem LeanRidgelet.lintegral_base_enorm_quadraticConstIteratedDeriv_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (κ : MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E))
  [κ.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) (j : ℕ)
  {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : Measurable (LeanRidgelet.quadraticConstIteratedDeriv j T)) :
  ∫⁻ (p : LeanRidgelet.QuadraticSymmetric E × E),
      ∫⁻ (t : ℝ),
        ‖iteratedDeriv j
              (LeanRidgelet.quadraticConstSlice (fun η ↦ T (g • η)) p)
              t‖ₑ ^
          2 ∂LeanRidgelet.quadraticBaseRelativeMeasure κ =
    ↑‖LinearMap.det ↑g.linear‖₊ *
      ∫⁻ (p : LeanRidgelet.QuadraticSymmetric E × E),
        ∫⁻ (t : ℝ),
          ‖iteratedDeriv j (LeanRidgelet.quadraticConstSlice T p) t‖ₑ ^
            2 ∂LeanRidgelet.quadraticBaseRelativeMeasure κ

theorem LeanRidgelet.lintegral_base_enorm_quadraticConstIteratedDeriv_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (κ :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticSymmetric E ×
        E))
  [κ.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  (j : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT :
    Measurable
      (LeanRidgelet.quadraticConstIteratedDeriv
        j T)) :
  ∫⁻ (p :
      LeanRidgelet.QuadraticSymmetric E ×
        E),
      ∫⁻ (t : ℝ),
        ‖iteratedDeriv j
              (LeanRidgelet.quadraticConstSlice
                (fun η ↦ T (g • η)) p)
              t‖ₑ ^
          2 ∂LeanRidgelet.quadraticBaseRelativeMeasure
        κ =
    ↑‖LinearMap.det ↑g.linear‖₊ *
      ∫⁻ (p :
        LeanRidgelet.QuadraticSymmetric
            E ×
          E),
        ∫⁻ (t : ℝ),
          ‖iteratedDeriv j
                (LeanRidgelet.quadraticConstSlice
                  T p)
                t‖ₑ ^
            2 ∂LeanRidgelet.quadraticBaseRelativeMeasure
          κ

**The `Γ`-seminorm identity through the factorization.**  Combining the previous result with
`LeanRidgelet.exists_map_prodAssoc_symm_quadraticRelativeMeasure_eq_smul`, the iterated integral
over the base parameter of the squared `L²` norm in the constant coefficient of the `j`-th
derivative — that is, the `j`-th seminorm of the informal space `L²(base; H^j(ℝ))` — is multiplied
by exactly `‖det L‖₊` under the pull-back.  The undetermined constant of the factorization cancels
between the two sides.

The additive Haar measure `lam` on the full parameter space does not appear in the conclusion.  It
is only a witness: the base parameter measure has no quasi-invariance of its own in this
development, so the statement is routed through the one that is proved upstairs.

This is what decides the design of the intermediate coefficient space. Since the action shifts the constant coefficient by an amount depending only on the other two, iterated differentiation in it commutes with the pull-back on the nose, and the norm of each derivative is unchanged slice by slice, by translation invariance of Lebesgue measure. Over the whole parameter measure the same identity holds with the constant that measure contributes on its own, the reciprocal of its Radon-Nikodym factor; combined with the factorization of the previous node it takes the iterated-integral shape a Sobolev space over the base would use, the factorization's own constant cancelling between the two sides.

So a Sobolev structure in the constant coefficient can be carried by the parameter representation, and the space it defines is the intermediate space through which the analysis and the synthesis are separately bounded — the point of the whole route, since such an operator need not be Hilbert-Schmidt. What this node establishes is that the structure is invariant; the two nodes below define the space and discharge the two measurability hypotheses carried here.

Lemma4.5.11

Statement uses 3

Statement dependency previews

Preview

Lemma 4.5.7

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

used by 1✓L∃∀N

An iterated derivative in one coordinate of a parameter is measurable in the others. Slice by slice this costs nothing; jointly in the parameter it is the continuity route, and for the analysis transform it is the derivative transfer.

Lean code for Lemma4.5.11●15 declarations

defdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

def LeanRidgelet.quadraticParameterPair.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (ξ : LeanRidgelet.QuadraticParameter E) :
  (LeanRidgelet.QuadraticSymmetric E × E) × ℝ

def LeanRidgelet.quadraticParameterPair.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  (LeanRidgelet.QuadraticSymmetric E ×
      E) ×
    ℝ

Implementation after :=

:=
  ((ξ.1, ξ.2.1), ξ.2.2)

**The quadratic parameter, reassociated.**  The parameter `(A, b, c)` read as the pair
`((A, b), c)` of the base parameter with the constant coefficient, so that the constant coefficient
is the last variable of a pair and the general statements above apply.  Reassociation of a product
is a linear homeomorphism, so this costs nothing.

defdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

def LeanRidgelet.quadraticConstPair.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (T : LeanRidgelet.QuadraticParameter E → ℂ) :
  (LeanRidgelet.QuadraticSymmetric E × E) × ℝ → ℂ

def LeanRidgelet.quadraticConstPair.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  (LeanRidgelet.QuadraticSymmetric E ×
        E) ×
      ℝ →
    ℂ

Implementation after :=

:=
  fun q ↦ T (q.1.1, q.1.2, q.2)

**A function of the quadratic parameter, reassociated.**  The same function read on pairs
`((A, b), c)`.  Its slices in the last variable are exactly the slices of
`LeanRidgelet.quadraticConstSlice`.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.continuous_quadraticParameterPair.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  Continuous LeanRidgelet.quadraticParameterPair

theorem LeanRidgelet.continuous_quadraticParameterPair.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  Continuous
    LeanRidgelet.quadraticParameterPair

The reassociation of the quadratic parameter is continuous.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.quadraticConstPair_slice.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (T : LeanRidgelet.QuadraticParameter E → ℂ)
  (p : LeanRidgelet.QuadraticSymmetric E × E) :
  (fun t ↦ LeanRidgelet.quadraticConstPair T (p, t)) =
    LeanRidgelet.quadraticConstSlice T p

theorem LeanRidgelet.quadraticConstPair_slice.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  (fun t ↦
      LeanRidgelet.quadraticConstPair T
        (p, t)) =
    LeanRidgelet.quadraticConstSlice T p

The slices of the reassociated function in the last variable are the slices in the constant
coefficient.  This holds by definition.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.quadraticConstIteratedDeriv_eq_parametric.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T : LeanRidgelet.QuadraticParameter E → ℂ) :
  LeanRidgelet.quadraticConstIteratedDeriv j T =
    MeasureTheory.parametricIteratedDeriv j
        (LeanRidgelet.quadraticConstPair T) ∘
      LeanRidgelet.quadraticParameterPair

theorem LeanRidgelet.quadraticConstIteratedDeriv_eq_parametric.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  LeanRidgelet.quadraticConstIteratedDeriv
      j T =
    MeasureTheory.parametricIteratedDeriv
        j
        (LeanRidgelet.quadraticConstPair
          T) ∘
      LeanRidgelet.quadraticParameterPair

**The derivative in the constant coefficient is the parametric derivative of the reassociated
function.**  This holds by definition, and it is what carries the general results to the quadratic
parameter space.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.iteratedDeriv_quadraticConstSlice_eq_parametric.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T : LeanRidgelet.QuadraticParameter E → ℂ)
  (p : LeanRidgelet.QuadraticSymmetric E × E) :
  iteratedDeriv j (LeanRidgelet.quadraticConstSlice T p) = fun t ↦
    MeasureTheory.parametricIteratedDeriv j
      (LeanRidgelet.quadraticConstPair T) (p, t)

theorem LeanRidgelet.iteratedDeriv_quadraticConstSlice_eq_parametric.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  iteratedDeriv j
      (LeanRidgelet.quadraticConstSlice T
        p) =
    fun t ↦
    MeasureTheory.parametricIteratedDeriv
      j
      (LeanRidgelet.quadraticConstPair T)
      (p, t)

The `j`-th derivative in the constant coefficient of a slice, as a slice of the parametric
iterated derivative.  This holds by definition.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.contDiff_quadraticConstPair.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {m : WithTop ℕ∞} {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ m T) :
  ContDiff ℝ m (LeanRidgelet.quadraticConstPair T)

theorem LeanRidgelet.contDiff_quadraticConstPair.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {m : WithTop ℕ∞}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ m T) :
  ContDiff ℝ m
    (LeanRidgelet.quadraticConstPair T)

Reassociating the quadratic parameter preserves joint smoothness, being a continuous linear
map.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.aestronglyMeasurable_iteratedDeriv_quadraticConstSlice_succ.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T : LeanRidgelet.QuadraticParameter E → ℂ)
  (p : LeanRidgelet.QuadraticSymmetric E × E) :
  MeasureTheory.AEStronglyMeasurable
    (iteratedDeriv (j + 1) (LeanRidgelet.quadraticConstSlice T p))
    MeasureTheory.volume

theorem LeanRidgelet.aestronglyMeasurable_iteratedDeriv_quadraticConstSlice_succ.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  MeasureTheory.AEStronglyMeasurable
    (iteratedDeriv (j + 1)
      (LeanRidgelet.quadraticConstSlice T
        p))
    MeasureTheory.volume

**The slice-wise hypothesis of
`LeanRidgelet.eLpNorm_iteratedDeriv_quadraticConstSlice_comp_smul` at a positive order, with no
hypothesis at all.** For every function on the quadratic parameter space, every base parameter and
every `j`, the `(j + 1)`-st derivative in the constant coefficient of the slice is almost everywhere
strongly measurable for Lebesgue measure on the line.  This is
`LeanRidgelet.stronglyMeasurable_iteratedDeriv_succ`; the caller instantiates the base parameter at
`LeanRidgelet.quadraticBaseLinearEquiv g p`.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.stronglyMeasurable_quadraticConstSlice.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  {T : LeanRidgelet.QuadraticParameter E → ℂ} (hT : Measurable T)
  (p : LeanRidgelet.QuadraticSymmetric E × E) :
  MeasureTheory.StronglyMeasurable
    (LeanRidgelet.quadraticConstSlice T p)

theorem LeanRidgelet.stronglyMeasurable_quadraticConstSlice.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : Measurable T)
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  MeasureTheory.StronglyMeasurable
    (LeanRidgelet.quadraticConstSlice T p)

A slice in the constant coefficient of a measurable function is measurable, the slice being the
composition with an affine map of the line.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.aestronglyMeasurable_iteratedDeriv_quadraticConstSlice.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)] (j : ℕ)
  {T : LeanRidgelet.QuadraticParameter E → ℂ} (hT : Measurable T)
  (p : LeanRidgelet.QuadraticSymmetric E × E) :
  MeasureTheory.AEStronglyMeasurable
    (iteratedDeriv j (LeanRidgelet.quadraticConstSlice T p))
    MeasureTheory.volume

theorem LeanRidgelet.aestronglyMeasurable_iteratedDeriv_quadraticConstSlice.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (j : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : Measurable T)
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  MeasureTheory.AEStronglyMeasurable
    (iteratedDeriv j
      (LeanRidgelet.quadraticConstSlice T
        p))
    MeasureTheory.volume

**The slice-wise hypothesis of
`LeanRidgelet.eLpNorm_iteratedDeriv_quadraticConstSlice_comp_smul` at every order.**  For a
measurable `T` the `j`-th derivative in the constant coefficient of every slice is almost everywhere
strongly measurable.  For a positive order the hypothesis on `T` is not used; it is needed only at
order `0`, where the iterated derivative is the slice itself.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.continuous_quadraticConstIteratedDeriv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (j : ℕ) {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑j) T) :
  Continuous (LeanRidgelet.quadraticConstIteratedDeriv j T)

theorem LeanRidgelet.continuous_quadraticConstIteratedDeriv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑j) T) :
  Continuous
    (LeanRidgelet.quadraticConstIteratedDeriv
      j T)

**The derivative in the constant coefficient of a jointly smooth function is jointly
continuous.**  The continuity route at the quadratic parameter space: for `T` that is `j` times
continuously differentiable on `QuadraticParameter E`, the function
`LeanRidgelet.quadraticConstIteratedDeriv j T` is continuous on the whole parameter space.  Only the
reassociation `LeanRidgelet.quadraticParameterPair` is interposed.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.measurable_quadraticConstIteratedDeriv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] (j : ℕ)
  {T : LeanRidgelet.QuadraticParameter E → ℂ} (hT : ContDiff ℝ (↑j) T) :
  Measurable (LeanRidgelet.quadraticConstIteratedDeriv j T)

theorem LeanRidgelet.measurable_quadraticConstIteratedDeriv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (j : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑j) T) :
  Measurable
    (LeanRidgelet.quadraticConstIteratedDeriv
      j T)

**The hypothesis `hT` of
`LeanRidgelet.lintegral_enorm_quadraticConstIteratedDeriv_comp_smul` and of
`LeanRidgelet.lintegral_base_enorm_quadraticConstIteratedDeriv_comp_smul`, from joint smoothness.**
For `T` that is `j` times continuously differentiable on the quadratic parameter space, the `j`-th
derivative in the constant coefficient is measurable as a function on that space.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.measurable_quadraticConstIteratedDeriv_succ.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] (j : ℕ)
  {T : LeanRidgelet.QuadraticParameter E → ℂ} (hT : ContDiff ℝ (↑j) T) :
  Measurable (LeanRidgelet.quadraticConstIteratedDeriv (j + 1) T)

theorem LeanRidgelet.measurable_quadraticConstIteratedDeriv_succ.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (j : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑j) T) :
  Measurable
    (LeanRidgelet.quadraticConstIteratedDeriv
      (j + 1) T)

**The same hypothesis one order up, from Mathlib's parametric derivative.**  `j` orders of joint
smoothness of `T` give measurability of the derivative of order `j + 1` in the constant coefficient,
because the last derivative only needs joint continuity of the previous one.  At `j = 0` this says
that joint continuity of `T` alone makes the first derivative in the constant coefficient
measurable.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.measurable_quadraticConstIteratedDeriv_of_eq.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)] (j : ℕ)
  {T S : LeanRidgelet.QuadraticParameter E → ℂ}
  (hTS :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E),
      iteratedDeriv j (LeanRidgelet.quadraticConstSlice T (A, b)) =
        fun c ↦ S (A, b, c))
  (hS : Measurable S) :
  Measurable (LeanRidgelet.quadraticConstIteratedDeriv j T)

theorem LeanRidgelet.measurable_quadraticConstIteratedDeriv_of_eq.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (j : ℕ)
  {T S :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hTS :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E),
      iteratedDeriv j
          (LeanRidgelet.quadraticConstSlice
            T (A, b)) =
        fun c ↦ S (A, b, c))
  (hS : Measurable S) :
  Measurable
    (LeanRidgelet.quadraticConstIteratedDeriv
      j T)

**The transfer route.**  If the `j`-th derivative in the constant coefficient of `T` is, base
parameter by base parameter, the slice of some function `S` on the parameter space, then
measurability of `S` gives measurability of the derivative.  This is a triviality, and it is stated
because it is exactly the shape that
`LeanRidgelet.iteratedDeriv_bochnerRidgelet_quadraticVectorFeature` produces: for the analysis
transform the derivative in the constant coefficient *is* another analysis transform, so no
smoothness in the first two coefficients has to be known.

theoremdefined in LeanRidgelet/HA/ParametricDerivMeasurable.lean

complete

theorem LeanRidgelet.measurable_quadraticConstIteratedDeriv_bochnerRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] {ρ : ℕ → ℝ → ℂ}
  (f : E → ℂ) (j : ℕ)
  {bound : LeanRidgelet.QuadraticSymmetric E × E → ℕ → E → ℝ}
  (hderiv : ∀ (i : ℕ) (z : ℝ), HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀ (i : ℕ) (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀ (i : ℕ) (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀ (i : ℕ) (A : LeanRidgelet.QuadraticSymmetric E) (b : E),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound (A, b) i x)
  (hboundint :
    ∀ (i : ℕ) (A : LeanRidgelet.QuadraticSymmetric E) (b : E),
      MeasureTheory.Integrable (bound (A, b) i) MeasureTheory.volume)
  (hρ : MeasureTheory.StronglyMeasurable (ρ j))
  (hf : MeasureTheory.AEStronglyMeasurable f MeasureTheory.volume) :
  Measurable
    (LeanRidgelet.quadraticConstIteratedDeriv j
      (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature (ρ 0)) f))

theorem LeanRidgelet.measurable_quadraticConstIteratedDeriv_bochnerRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ) (j : ℕ)
  {bound :
    LeanRidgelet.QuadraticSymmetric E ×
        E →
      ℕ → E → ℝ}
  (hderiv :
    ∀ (i : ℕ) (z : ℝ),
      HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀ (i : ℕ)
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀ (i : ℕ)
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀ (i : ℕ)
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature
                    (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound (A, b) i x)
  (hboundint :
    ∀ (i : ℕ)
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E),
      MeasureTheory.Integrable
        (bound (A, b) i)
        MeasureTheory.volume)
  (hρ :
    MeasureTheory.StronglyMeasurable
      (ρ j))
  (hf :
    MeasureTheory.AEStronglyMeasurable f
      MeasureTheory.volume) :
  Measurable
    (LeanRidgelet.quadraticConstIteratedDeriv
      j
      (LeanRidgelet.bochnerRidgelet
        MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature
          (ρ 0))
        f))

**The hypothesis `hT` of
`LeanRidgelet.lintegral_enorm_quadraticConstIteratedDeriv_comp_smul` for the analysis transform.**
Under the hypotheses of `LeanRidgelet.iteratedDeriv_bochnerRidgelet_quadraticVectorFeature`, imposed
at every base parameter, the `j`-th derivative in the constant coefficient of the analysis transform
of the quadratic feature of `ρ 0` is measurable on the parameter space, because it equals the
analysis transform of the quadratic feature of `ρ j`, which is measurable by
`LeanRidgelet.stronglyMeasurable_bochnerRidgelet_quadraticVectorFeature`.

This is the instantiation to use in practice.  Nothing is assumed about differentiability of the
transform in the symmetric or the linear coefficient -- only in the constant one, where the transfer
lives.

The two hypotheses the shear node carries are of very different difficulty, and only one of them has content. The slice-wise one is free: a derivative is measurable for every function, differentiable or not, because it vanishes off the differentiability set and that set is Borel, so iterating costs nothing and a positive order needs no hypothesis at all.

The one in the parameter has content, and there are two routes. Mathlib's parametric-derivative lemma is sharp for one step — joint continuity in, joint measurability out, with no differentiability assumed — but it cannot be iterated, because the continuity it consumes is not what it produces. The continuity route supplies that: the parametric derivative of a jointly smooth function is again jointly smooth, one order down, so an induction gives joint continuity of the parametric iterated derivative, and feeding that back into the one-step lemma gains an order. Reassociating the three coefficients of a quadratic parameter is a linear homeomorphism, so the general statements apply there unchanged.

Neither route sees the analysis transform, whose joint smoothness in the parameter is not known here. For it there is a third route, and it is the cheap one: its derivative in the constant coefficient is another analysis transform, by the transfer, and that one is measurable in the parameter by the Bochner measurability node. So for the one function the argument needs, the hypothesis is discharged from the transfer hypotheses alone.

Lemma4.5.12

Statement uses 4

Statement dependency previews

Preview

Lemma 4.5.9

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

Used by 2

Reverse dependency previews

Preview

Lemma 4.5.13

Loading preview

Reverse dependency preview content is loaded from the rendered-fragment cache.

✓L∃∀N

The intermediate coefficient space. Order-k Sobolev in the constant coefficient of the parameter, over the parameter measure: the action scales its seminorm by the square root of the determinant of the linear part, the analysis transform's seminorm is computed by the features, and the synthesis integral is dominated by the seminorm.

Lean code for Lemma4.5.12●23 declarations

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.quadraticConstIteratedDeriv_zero.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (T : LeanRidgelet.QuadraticParameter E → ℂ) :
  LeanRidgelet.quadraticConstIteratedDeriv 0 T = T

theorem LeanRidgelet.quadraticConstIteratedDeriv_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  LeanRidgelet.quadraticConstIteratedDeriv
      0 T =
    T

At order `0` the derivative in the constant coefficient is the function itself.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.quadraticConstIteratedDeriv_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (j : ℕ) (T : LeanRidgelet.QuadraticParameter E → ℂ)
  (p : LeanRidgelet.QuadraticSymmetric E × E) (t : ℝ) :
  LeanRidgelet.quadraticConstIteratedDeriv j T (p.1, p.2, t) =
    iteratedDeriv j (LeanRidgelet.quadraticConstSlice T p) t

theorem LeanRidgelet.quadraticConstIteratedDeriv_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (j : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (p :
    LeanRidgelet.QuadraticSymmetric E × E)
  (t : ℝ) :
  LeanRidgelet.quadraticConstIteratedDeriv
      j T (p.1, p.2, t) =
    iteratedDeriv j
      (LeanRidgelet.quadraticConstSlice T
        p)
      t

The derivative in the constant coefficient evaluated at a split parameter is the derivative of
the slice at the base parameter.  This is the bridge between the global form of the seminorm and its
base-level form.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.quadraticConstIteratedDeriv_bochnerRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ)
  {bound : LeanRidgelet.QuadraticSymmetric E → E → ℕ → E → ℝ}
  (hderiv : ∀ (i : ℕ) (z : ℝ), HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound A b i x)
  (hboundint :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ),
      MeasureTheory.Integrable (bound A b i) MeasureTheory.volume)
  (j : ℕ) :
  LeanRidgelet.quadraticConstIteratedDeriv j
      (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature (ρ 0)) f) =
    LeanRidgelet.bochnerRidgelet MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature (ρ j)) f

theorem LeanRidgelet.quadraticConstIteratedDeriv_bochnerRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ)
  {bound :
    LeanRidgelet.QuadraticSymmetric E →
      E → ℕ → E → ℝ}
  (hderiv :
    ∀ (i : ℕ) (z : ℝ),
      HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature
                    (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound A b i x)
  (hboundint :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ),
      MeasureTheory.Integrable
        (bound A b i)
        MeasureTheory.volume)
  (j : ℕ) :
  LeanRidgelet.quadraticConstIteratedDeriv
      j
      (LeanRidgelet.bochnerRidgelet
        MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature
          (ρ 0))
        f) =
    LeanRidgelet.bochnerRidgelet
      MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature
        (ρ j))
      f

**The derivative transfer, on the whole parameter space.**  For a sequence of analysis features
each the derivative of the previous one, the `j`-th derivative in the constant coefficient of the
analysis transform of the first feature is the analysis transform of the `j`-th feature, at every
parameter.  This is `LeanRidgelet.iteratedDeriv_bochnerRidgelet_quadraticVectorFeature` with its
hypotheses taken uniformly in the base parameter, so that the conclusion is an identity of two
functions on the parameter space rather than one for each frozen slice.

defdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

def LeanRidgelet.quadraticSobolevSeminorm.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) (T : LeanRidgelet.QuadraticParameter E → ℂ) : ENNReal

def LeanRidgelet.quadraticSobolevSeminorm.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  ENNReal

Implementation after :=

:=
  ∑ j ∈ Finset.range (k + 1),
    eLpNorm (quadraticConstIteratedDeriv j T) 2 (quadraticRelativeMeasure lam)

**The order-`k` Sobolev seminorm in the constant coefficient.**  The sum over `j ≤ k` of the
`L²` norms, against the relatively invariant parameter measure, of the `j`-th derivative in the
constant coefficient.  This is the seminorm of the intermediate coefficient space `Γ^k`, informally
`L²(base ; H^k(ℝ))`; see `LeanRidgelet.quadraticBaseSobolevSeminorm` for the reading that makes the
informal description literal.  It is `ℝ≥0∞`-valued, so it is defined for every `T` with no
finiteness or measurability side condition.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.quadraticSobolevSeminorm_zero.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (T : LeanRidgelet.QuadraticParameter E → ℂ) :
  LeanRidgelet.quadraticSobolevSeminorm lam 0 T =
    MeasureTheory.eLpNorm T 2
      (LeanRidgelet.quadraticRelativeMeasure lam)

theorem LeanRidgelet.quadraticSobolevSeminorm_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  LeanRidgelet.quadraticSobolevSeminorm
      lam 0 T =
    MeasureTheory.eLpNorm T 2
      (LeanRidgelet.quadraticRelativeMeasure
        lam)

At order `0` the seminorm is the `L²` norm of the coefficient function itself.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.eLpNorm_quadraticConstIteratedDeriv_le_quadraticSobolevSeminorm.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  {j k : ℕ} (hj : j ≤ k) (T : LeanRidgelet.QuadraticParameter E → ℂ) :
  MeasureTheory.eLpNorm (LeanRidgelet.quadraticConstIteratedDeriv j T) 2
      (LeanRidgelet.quadraticRelativeMeasure lam) ≤
    LeanRidgelet.quadraticSobolevSeminorm lam k T

theorem LeanRidgelet.eLpNorm_quadraticConstIteratedDeriv_le_quadraticSobolevSeminorm.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  {j k : ℕ} (hj : j ≤ k)
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  MeasureTheory.eLpNorm
      (LeanRidgelet.quadraticConstIteratedDeriv
        j T)
      2
      (LeanRidgelet.quadraticRelativeMeasure
        lam) ≤
    LeanRidgelet.quadraticSobolevSeminorm
      lam k T

Each term of the seminorm is bounded by the seminorm.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.eLpNorm_le_quadraticSobolevSeminorm.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) (T : LeanRidgelet.QuadraticParameter E → ℂ) :
  MeasureTheory.eLpNorm T 2
      (LeanRidgelet.quadraticRelativeMeasure lam) ≤
    LeanRidgelet.quadraticSobolevSeminorm lam k T

theorem LeanRidgelet.eLpNorm_le_quadraticSobolevSeminorm.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  MeasureTheory.eLpNorm T 2
      (LeanRidgelet.quadraticRelativeMeasure
        lam) ≤
    LeanRidgelet.quadraticSobolevSeminorm
      lam k T

The seminorm dominates the plain `L²` norm: the space sits inside `L²` of the parameter
measure.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.quadraticSobolevSeminorm_mono.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  {k l : ℕ} (h : k ≤ l) (T : LeanRidgelet.QuadraticParameter E → ℂ) :
  LeanRidgelet.quadraticSobolevSeminorm lam k T ≤
    LeanRidgelet.quadraticSobolevSeminorm lam l T

theorem LeanRidgelet.quadraticSobolevSeminorm_mono.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  {k l : ℕ} (h : k ≤ l)
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  LeanRidgelet.quadraticSobolevSeminorm
      lam k T ≤
    LeanRidgelet.quadraticSobolevSeminorm
      lam l T

The seminorm increases with the order, so the spaces decrease.

structure(2 fields)defined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

structure LeanRidgelet.MemQuadraticSobolev.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) (T : LeanRidgelet.QuadraticParameter E → ℂ) : Prop

structure LeanRidgelet.MemQuadraticSobolev.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  Prop

**Membership in the intermediate coefficient space `Γ^k`.**  A coefficient function lies in the
space when each of its derivatives up to order `k` in the constant coefficient is measurable as a
function on the parameter space and the order-`k` Sobolev seminorm is finite.  The measurability is
carried in exactly the shape `LeanRidgelet.HA.QuadraticShear` uses it; it is not available anywhere
in this development for `iteratedDeriv` in a parameter, and is what the statements below consume.

Fields

measurable : ∀ j ≤ k, Measurable (LeanRidgelet.quadraticConstIteratedDeriv j T)

Every derivative up to order `k` in the constant coefficient is measurable.

seminorm_lt_top : LeanRidgelet.quadraticSobolevSeminorm lam k T < ⊤

The order-`k` Sobolev seminorm is finite.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.MemQuadraticSobolev.memLp.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  {lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)}
  {k : ℕ} {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (h : LeanRidgelet.MemQuadraticSobolev lam k T) :
  MeasureTheory.MemLp T 2 (LeanRidgelet.quadraticRelativeMeasure lam)

theorem LeanRidgelet.MemQuadraticSobolev.memLp.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E)}
  {k : ℕ}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (h :
    LeanRidgelet.MemQuadraticSobolev lam k
      T) :
  MeasureTheory.MemLp T 2
    (LeanRidgelet.quadraticRelativeMeasure
      lam)

A member of the space is square integrable for the parameter measure.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.quadraticSobolevSeminorm_comp_smul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) (k : ℕ)
  {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT :
    ∀ j ≤ k,
      Measurable (LeanRidgelet.quadraticConstIteratedDeriv j T)) :
  (LeanRidgelet.quadraticSobolevSeminorm lam k fun η ↦ T (g • η)) =
    ↑‖LinearMap.det ↑g.linear‖₊ ^ (1 / 2) *
      LeanRidgelet.quadraticSobolevSeminorm lam k T

theorem LeanRidgelet.quadraticSobolevSeminorm_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  (k : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT :
    ∀ j ≤ k,
      Measurable
        (LeanRidgelet.quadraticConstIteratedDeriv
          j T)) :
  (LeanRidgelet.quadraticSobolevSeminorm
      lam k fun η ↦ T (g • η)) =
    ↑‖LinearMap.det ↑g.linear‖₊ ^
        (1 / 2) *
      LeanRidgelet.quadraticSobolevSeminorm
        lam k T

**The action scales the seminorm by a known factor.**  Pulling a coefficient function back along
the parameter action of `g` multiplies its order-`k` Sobolev seminorm in the constant coefficient by
exactly `‖det L‖₊ ^ (1 / 2)`, the square root of the constant that the quasi-invariance of
`LeanRidgelet.quadraticRelativeMeasure` contributes; the square root appears because the seminorm is
an `L²` norm and the quasi-invariance is a statement about the squared integrand.

Nothing else happens, and that is the point: by
`LeanRidgelet.quadraticConstIteratedDeriv_comp_smul` the action is a shear in the constant
coefficient, so it neither differentiates nor rescales that coordinate, and the derivative of a
pull-back is the pull-back of the derivative with no factor.  The whole Sobolev structure is
therefore carried along, term by term, with one and the same constant, which is why the constant
factors out of the sum.  In particular the same square-root normalization of the pull-back that
makes `LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation` unitary makes this seminorm
invariant on the nose.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.MemQuadraticSobolev.comp_smul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  {lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)}
  [lam.IsAddHaarMeasure] {k : ℕ}
  {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (h : LeanRidgelet.MemQuadraticSobolev lam k T) (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.MemQuadraticSobolev lam k fun η ↦ T (g • η)

theorem LeanRidgelet.MemQuadraticSobolev.comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E)}
  [lam.IsAddHaarMeasure] {k : ℕ}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (h :
    LeanRidgelet.MemQuadraticSobolev lam k
      T)
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.MemQuadraticSobolev lam k
    fun η ↦ T (g • η)

**The predicate is preserved by the action.**  Pulling back along the parameter action of `g`
keeps a coefficient function in the space: the measurability is transported by the equivariance of
the derivative in the constant coefficient, and the seminorm stays finite because the action only
multiplies it by the finite constant of
`LeanRidgelet.quadraticSobolevSeminorm_comp_smul`.  Together with that identity this is what makes
`Γ^k` a legitimate intermediate space for the reconstruction argument.

defdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

def LeanRidgelet.quadraticBaseSobolevSeminorm.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (κ : MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E))
  (k : ℕ) (T : LeanRidgelet.QuadraticParameter E → ℂ) : ENNReal

def LeanRidgelet.quadraticBaseSobolevSeminorm.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (κ :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticSymmetric E ×
        E))
  (k : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  ENNReal

Implementation after :=

:=
  ∑ j ∈ Finset.range (k + 1),
    (∫⁻ p : QuadraticSymmetric E × E,
        eLpNorm (iteratedDeriv j (quadraticConstSlice T p)) 2 (volume : Measure ℝ) ^ 2
        ∂(quadraticBaseRelativeMeasure κ)) ^ ((1 : ℝ) / 2)

**The seminorm read through the factorization of the parameter measure.**  The order-`k` Sobolev
seminorm of a coefficient function, with the constant coefficient integrated first: the sum over
`j ≤ k` of the square roots of the integrals over the base parameter of the squared `L²` norm on the
line of the `j`-th derivative of the slice.  This is the literal form of the informal description
`Γ^k = L²(base ; H^k(ℝ))`, and by
`LeanRidgelet.exists_quadraticSobolevSeminorm_eq_mul_quadraticBaseSobolevSeminorm` it agrees with
`LeanRidgelet.quadraticSobolevSeminorm` up to the positive finite constant of the factorization.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.quadraticBaseSobolevSeminorm_eq_lintegral.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (κ : MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E))
  (k : ℕ) (T : LeanRidgelet.QuadraticParameter E → ℂ) :
  LeanRidgelet.quadraticBaseSobolevSeminorm κ k T =
    ∑ j ∈ Finset.range (k + 1),
      (∫⁻ (p : LeanRidgelet.QuadraticSymmetric E × E),
          ∫⁻ (t : ℝ),
            ‖iteratedDeriv j (LeanRidgelet.quadraticConstSlice T p)
                  t‖ₑ ^
              2 ∂LeanRidgelet.quadraticBaseRelativeMeasure κ) ^
        (1 / 2)

theorem LeanRidgelet.quadraticBaseSobolevSeminorm_eq_lintegral.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (κ :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticSymmetric E ×
        E))
  (k : ℕ)
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  LeanRidgelet.quadraticBaseSobolevSeminorm
      κ k T =
    ∑ j ∈ Finset.range (k + 1),
      (∫⁻ (p :
          LeanRidgelet.QuadraticSymmetric
              E ×
            E),
          ∫⁻ (t : ℝ),
            ‖iteratedDeriv j
                  (LeanRidgelet.quadraticConstSlice
                    T p)
                  t‖ₑ ^
              2 ∂LeanRidgelet.quadraticBaseRelativeMeasure
            κ) ^
        (1 / 2)

The base-level seminorm as an iterated lower Lebesgue integral, which is the form the invariance
statements of `LeanRidgelet.HA.QuadraticShear` are stated in.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.quadraticBaseSobolevSeminorm_comp_smul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (κ : MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E))
  [κ.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) (k : ℕ)
  {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT :
    ∀ j ≤ k,
      Measurable (LeanRidgelet.quadraticConstIteratedDeriv j T)) :
  (LeanRidgelet.quadraticBaseSobolevSeminorm κ k fun η ↦ T (g • η)) =
    ↑‖LinearMap.det ↑g.linear‖₊ ^ (1 / 2) *
      LeanRidgelet.quadraticBaseSobolevSeminorm κ k T

theorem LeanRidgelet.quadraticBaseSobolevSeminorm_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (κ :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticSymmetric E ×
        E))
  [κ.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  (k : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT :
    ∀ j ≤ k,
      Measurable
        (LeanRidgelet.quadraticConstIteratedDeriv
          j T)) :
  (LeanRidgelet.quadraticBaseSobolevSeminorm
      κ k fun η ↦ T (g • η)) =
    ↑‖LinearMap.det ↑g.linear‖₊ ^
        (1 / 2) *
      LeanRidgelet.quadraticBaseSobolevSeminorm
        κ k T

**Invariance of the base-level seminorm.**  The same constant `‖det L‖₊ ^ (1 / 2)` appears one
factor down, after the parameter measure has been factored over the constant coefficient; the
undetermined constant of the factorization cancels between the two sides.  As in
`LeanRidgelet.lintegral_base_enorm_quadraticConstIteratedDeriv_comp_smul`, the additive Haar measure
`lam` on the full parameter space is only a witness: it does not appear in the conclusion, and it is
there because the base parameter measure has no quasi-invariance of its own in this development.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.exists_quadraticSobolevSeminorm_eq_mul_quadraticBaseSobolevSeminorm.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (κ : MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E))
  [κ.IsAddHaarMeasure] (k : ℕ)
  {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT :
    ∀ j ≤ k,
      Measurable (LeanRidgelet.quadraticConstIteratedDeriv j T)) :
  ∃ c,
    c ≠ 0 ∧
      c ≠ ⊤ ∧
        LeanRidgelet.quadraticSobolevSeminorm lam k T =
          c ^ (1 / 2) * LeanRidgelet.quadraticBaseSobolevSeminorm κ k T

theorem LeanRidgelet.exists_quadraticSobolevSeminorm_eq_mul_quadraticBaseSobolevSeminorm.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (κ :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticSymmetric E ×
        E))
  [κ.IsAddHaarMeasure] (k : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT :
    ∀ j ≤ k,
      Measurable
        (LeanRidgelet.quadraticConstIteratedDeriv
          j T)) :
  ∃ c,
    c ≠ 0 ∧
      c ≠ ⊤ ∧
        LeanRidgelet.quadraticSobolevSeminorm
            lam k T =
          c ^ (1 / 2) *
            LeanRidgelet.quadraticBaseSobolevSeminorm
              κ k T

**The two readings of the seminorm agree.**  The order-`k` Sobolev seminorm over the parameter
measure is a positive finite multiple of its base-level form, the multiple being the square root of
the constant of the factorization of `LeanRidgelet.HA.QuadraticParameterFactor`.  This is the
precise content of the informal identification `Γ^k = L²(base ; H^k(ℝ))`; the constant is
undetermined because `lam` and `κ` are each only fixed up to a scalar.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.quadraticSobolevSeminorm_bochnerRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ)
  {bound : LeanRidgelet.QuadraticSymmetric E → E → ℕ → E → ℝ}
  (hderiv : ∀ (i : ℕ) (z : ℝ), HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound A b i x)
  (hboundint :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ),
      MeasureTheory.Integrable (bound A b i) MeasureTheory.volume)
  (k : ℕ) :
  LeanRidgelet.quadraticSobolevSeminorm lam k
      (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature (ρ 0)) f) =
    ∑ j ∈ Finset.range (k + 1),
      MeasureTheory.eLpNorm
        (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
          (LeanRidgelet.quadraticVectorFeature (ρ j)) f)
        2 (LeanRidgelet.quadraticRelativeMeasure lam)

theorem LeanRidgelet.quadraticSobolevSeminorm_bochnerRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ)
  {bound :
    LeanRidgelet.QuadraticSymmetric E →
      E → ℕ → E → ℝ}
  (hderiv :
    ∀ (i : ℕ) (z : ℝ),
      HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature
                    (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound A b i x)
  (hboundint :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ),
      MeasureTheory.Integrable
        (bound A b i)
        MeasureTheory.volume)
  (k : ℕ) :
  LeanRidgelet.quadraticSobolevSeminorm
      lam k
      (LeanRidgelet.bochnerRidgelet
        MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature
          (ρ 0))
        f) =
    ∑ j ∈ Finset.range (k + 1),
      MeasureTheory.eLpNorm
        (LeanRidgelet.bochnerRidgelet
          MeasureTheory.volume
          (LeanRidgelet.quadraticVectorFeature
            (ρ j))
          f)
        2
        (LeanRidgelet.quadraticRelativeMeasure
          lam)

**The seminorm of the analysis transform is computed by the features.**  For a sequence of
analysis features each the derivative of the previous one, the order-`k` Sobolev seminorm in the
constant coefficient of the analysis transform of the first feature is the sum over `j ≤ k` of the
`L²` norms of the analysis transforms of the `j`-th features.

This is the payoff of putting the Sobolev structure in the constant coefficient: no Fourier
transform appears, and the smoothness of the analysis feature is exactly what puts its transform in
the space.  Compare `LeanRidgelet.eLpNorm_pow_smul_angularFourier_bochnerRidgelet_slice`, which
records the same information one slice at a time and on the Fourier side, at the cost of a
Plancherel constant.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.memQuadraticSobolev_bochnerRidgelet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ)
  {bound : LeanRidgelet.QuadraticSymmetric E → E → ℕ → E → ℝ}
  (hderiv : ∀ (i : ℕ) (z : ℝ), HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound A b i x)
  (hboundint :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ),
      MeasureTheory.Integrable (bound A b i) MeasureTheory.volume)
  (k : ℕ)
  (htmeas :
    ∀ j ≤ k,
      Measurable
        (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
          (LeanRidgelet.quadraticVectorFeature (ρ j)) f))
  (htfin :
    ∀ j ≤ k,
      MeasureTheory.eLpNorm
          (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature (ρ j)) f)
          2 (LeanRidgelet.quadraticRelativeMeasure lam) <
        ⊤) :
  LeanRidgelet.MemQuadraticSobolev lam k
    (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature (ρ 0)) f)

theorem LeanRidgelet.memQuadraticSobolev_bochnerRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ)
  {bound :
    LeanRidgelet.QuadraticSymmetric E →
      E → ℕ → E → ℝ}
  (hderiv :
    ∀ (i : ℕ) (z : ℝ),
      HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature
                    (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound A b i x)
  (hboundint :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ),
      MeasureTheory.Integrable
        (bound A b i)
        MeasureTheory.volume)
  (k : ℕ)
  (htmeas :
    ∀ j ≤ k,
      Measurable
        (LeanRidgelet.bochnerRidgelet
          MeasureTheory.volume
          (LeanRidgelet.quadraticVectorFeature
            (ρ j))
          f))
  (htfin :
    ∀ j ≤ k,
      MeasureTheory.eLpNorm
          (LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              (ρ j))
            f)
          2
          (LeanRidgelet.quadraticRelativeMeasure
            lam) <
        ⊤) :
  LeanRidgelet.MemQuadraticSobolev lam k
    (LeanRidgelet.bochnerRidgelet
      MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature
        (ρ 0))
      f)

**Smoothness of the analysis feature puts its transform in the space.**  If in addition each of
the analysis transforms of the features up to order `k` is measurable in the parameter and square
integrable for the parameter measure, then the analysis transform of the first feature lies in
`Γ^k`.  The two extra hypotheses are conditions on the features, not on the transform of the first
one, which is the form a concrete ridgelet function is checked against.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.enorm_bochnerSynthesis_le_quadraticSobolevSeminorm_mul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) (φ : ℝ → ℂ) {γ : LeanRidgelet.QuadraticParameter E → ℂ}
  (x : E)
  (hγ : AEMeasurable γ (LeanRidgelet.quadraticRelativeMeasure lam))
  (hφ :
    AEMeasurable (fun ξ ↦ LeanRidgelet.quadraticVectorFeature φ x ξ)
      (LeanRidgelet.quadraticRelativeMeasure lam)) :
  ‖LeanRidgelet.bochnerSynthesis
        (LeanRidgelet.quadraticRelativeMeasure lam)
        (LeanRidgelet.quadraticVectorFeature φ) γ x‖ₑ ≤
    LeanRidgelet.quadraticSobolevSeminorm lam k γ *
      MeasureTheory.eLpNorm
        (fun ξ ↦ LeanRidgelet.quadraticVectorFeature φ x ξ) 2
        (LeanRidgelet.quadraticRelativeMeasure lam)

theorem LeanRidgelet.enorm_bochnerSynthesis_le_quadraticSobolevSeminorm_mul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) (φ : ℝ → ℂ)
  {γ :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (x : E)
  (hγ :
    AEMeasurable γ
      (LeanRidgelet.quadraticRelativeMeasure
        lam))
  (hφ :
    AEMeasurable
      (fun ξ ↦
        LeanRidgelet.quadraticVectorFeature
          φ x ξ)
      (LeanRidgelet.quadraticRelativeMeasure
        lam)) :
  ‖LeanRidgelet.bochnerSynthesis
        (LeanRidgelet.quadraticRelativeMeasure
          lam)
        (LeanRidgelet.quadraticVectorFeature
          φ)
        γ x‖ₑ ≤
    LeanRidgelet.quadraticSobolevSeminorm
        lam k γ *
      MeasureTheory.eLpNorm
        (fun ξ ↦
          LeanRidgelet.quadraticVectorFeature
            φ x ξ)
        2
        (LeanRidgelet.quadraticRelativeMeasure
          lam)

**The synthesis integral is dominated by the seminorm.**  The Bochner synthesis of a coefficient
function against the quadratic synthesis feature is bounded, at every data point, by the order-`k`
Sobolev seminorm of the coefficient function times the `L²` norm in the parameter of the synthesis
feature at that point.  This is the dual form of the synthesis bound: the second factor is the dual
quantity for the synthesis feature, and it is the only place the feature enters.

Only the order-`0` term of the seminorm is used, so the bound holds for every order; the smoothness
in the constant coefficient is what the *analysis* side needs, not the synthesis side.  Combining
this with `LeanRidgelet.quadraticSobolevSeminorm_bochnerRidgelet` is how a bound on the composite is
assembled without either operator being Hilbert--Schmidt, but that assembly is not carried out
here.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.measurable_quadraticConstIteratedDeriv_le_of_contDiff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)] {k : ℕ}
  {T : LeanRidgelet.QuadraticParameter E → ℂ} (hT : ContDiff ℝ (↑k) T)
  (j : ℕ) :
  j ≤ k → Measurable (LeanRidgelet.quadraticConstIteratedDeriv j T)

theorem LeanRidgelet.measurable_quadraticConstIteratedDeriv_le_of_contDiff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {k : ℕ}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑k) T) (j : ℕ) :
  j ≤ k →
    Measurable
      (LeanRidgelet.quadraticConstIteratedDeriv
        j T)

The derivatives up to order `k` in the constant coefficient of a `k` times continuously
differentiable coefficient function are measurable on the parameter space.  This is the continuity
route of `LeanRidgelet.HA.ParametricDerivMeasurable`, packaged in the shape the statements above
consume, so that for a smooth coefficient function nothing has to be assumed.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.quadraticSobolevSeminorm_comp_smul_of_contDiff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) {k : ℕ}
  {T : LeanRidgelet.QuadraticParameter E → ℂ} (hT : ContDiff ℝ (↑k) T) :
  (LeanRidgelet.quadraticSobolevSeminorm lam k fun η ↦ T (g • η)) =
    ↑‖LinearMap.det ↑g.linear‖₊ ^ (1 / 2) *
      LeanRidgelet.quadraticSobolevSeminorm lam k T

theorem LeanRidgelet.quadraticSobolevSeminorm_comp_smul_of_contDiff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  {k : ℕ}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑k) T) :
  (LeanRidgelet.quadraticSobolevSeminorm
      lam k fun η ↦ T (g • η)) =
    ↑‖LinearMap.det ↑g.linear‖₊ ^
        (1 / 2) *
      LeanRidgelet.quadraticSobolevSeminorm
        lam k T

**The invariance of the seminorm for a smooth coefficient function, with no measurability
hypothesis left.**  For `T` that is `k` times continuously differentiable on the parameter space the
order-`k` seminorm of the pull-back is the square root of the determinant of the linear part times
the seminorm of `T`.  This is `LeanRidgelet.quadraticSobolevSeminorm_comp_smul` with its
measurability family discharged.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.memQuadraticSobolev_of_contDiff.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  {lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)}
  {k : ℕ} {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑k) T)
  (hfin : LeanRidgelet.quadraticSobolevSeminorm lam k T < ⊤) :
  LeanRidgelet.MemQuadraticSobolev lam k T

theorem LeanRidgelet.memQuadraticSobolev_of_contDiff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E)}
  {k : ℕ}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑k) T)
  (hfin :
    LeanRidgelet.quadraticSobolevSeminorm
        lam k T <
      ⊤) :
  LeanRidgelet.MemQuadraticSobolev lam k T

**Membership from smoothness and finiteness alone.**  A `k` times continuously differentiable
coefficient function of finite order-`k` seminorm lies in the space.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevSpace.lean

complete

theorem LeanRidgelet.memQuadraticSobolev_bochnerRidgelet_of_stronglyMeasurable.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ)
  {bound : LeanRidgelet.QuadraticSymmetric E → E → ℕ → E → ℝ}
  (hderiv : ∀ (i : ℕ) (z : ℝ), HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound A b i x)
  (hboundint :
    ∀ (A : LeanRidgelet.QuadraticSymmetric E) (b : E) (i : ℕ),
      MeasureTheory.Integrable (bound A b i) MeasureTheory.volume)
  (k : ℕ) (hρmeas : ∀ (i : ℕ), MeasureTheory.StronglyMeasurable (ρ i))
  (hf : MeasureTheory.AEStronglyMeasurable f MeasureTheory.volume)
  (htfin :
    ∀ j ≤ k,
      MeasureTheory.eLpNorm
          (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature (ρ j)) f)
          2 (LeanRidgelet.quadraticRelativeMeasure lam) <
        ⊤) :
  LeanRidgelet.MemQuadraticSobolev lam k
    (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature (ρ 0)) f)

theorem LeanRidgelet.memQuadraticSobolev_bochnerRidgelet_of_stronglyMeasurable.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  {ρ : ℕ → ℝ → ℂ} (f : E → ℂ)
  {bound :
    LeanRidgelet.QuadraticSymmetric E →
      E → ℕ → E → ℝ}
  (hderiv :
    ∀ (i : ℕ) (z : ℝ),
      HasDerivAt (ρ i) (ρ (i + 1) z) z)
  (hmeas :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.AEStronglyMeasurable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hint :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ) (c : ℝ),
      MeasureTheory.Integrable
        (fun x ↦
          f x *
            (starRingEnd ℂ)
              (LeanRidgelet.quadraticVectorFeature
                (ρ i) x (A, b, c)))
        MeasureTheory.volume)
  (hbound :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ),
      ∀ᵐ (x : E),
        ∀ (c : ℝ),
          ‖f x *
                (starRingEnd ℂ)
                  (LeanRidgelet.quadraticVectorFeature
                    (ρ (i + 1)) x
                    (A, b, c))‖ ≤
            bound A b i x)
  (hboundint :
    ∀
      (A :
        LeanRidgelet.QuadraticSymmetric E)
      (b : E) (i : ℕ),
      MeasureTheory.Integrable
        (bound A b i)
        MeasureTheory.volume)
  (k : ℕ)
  (hρmeas :
    ∀ (i : ℕ),
      MeasureTheory.StronglyMeasurable
        (ρ i))
  (hf :
    MeasureTheory.AEStronglyMeasurable f
      MeasureTheory.volume)
  (htfin :
    ∀ j ≤ k,
      MeasureTheory.eLpNorm
          (LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              (ρ j))
            f)
          2
          (LeanRidgelet.quadraticRelativeMeasure
            lam) <
        ⊤) :
  LeanRidgelet.MemQuadraticSobolev lam k
    (LeanRidgelet.bochnerRidgelet
      MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature
        (ρ 0))
      f)

**The analysis transform lies in the space, with only checkable hypotheses.**  This is
`LeanRidgelet.memQuadraticSobolev_bochnerRidgelet` with its measurability hypothesis on the
transforms replaced by measurability of the analysis features and of the data, which is what
`LeanRidgelet.stronglyMeasurable_bochnerRidgelet_quadraticVectorFeature` supplies.  What is left
assumed is the transfer of derivatives and finiteness of each transform's `L²` norm; no
measurability of an `iteratedDeriv` in a parameter is assumed anywhere.

This is the space the boundedness route needs, and the seminorm is the sum over orders up to k of the parameter-L² norms of the derivatives in the constant coefficient. It is valued in the extended nonnegative reals, so it is defined for every coefficient function with no side condition, and membership is finiteness of it together with the measurability the previous node supplies. Two readings agree, up to the constant of the factorization: the global one over the parameter measure and the iterated one over the base, which is what makes the informal description a Sobolev-valued L² over the base literal.

Three statements make it an intermediate space rather than a definition. The action carries it, by the shear: the seminorm of a pull-back is the square root of the determinant of the linear part times the seminorm, at every order, so membership is preserved and the action is bounded on it with a known factor. The analysis side is an identity, not an estimate: the seminorm of the transform of the first feature is the sum of the parameter-L² norms of the transforms of the successive derivative features, so smoothness of the analysis feature is exactly what puts its transform in the space, and no Fourier transform appears in the statement. The synthesis side is Cauchy-Schwarz in the parameter, in dual form: the synthesis integral against a coefficient function is bounded at each data point by the seminorm times the parameter-L² norm of the synthesis feature there, and only the order-zero term of the seminorm is used, so the smoothness is spent entirely on the analysis side.

What is deliberately not done is the Hilbert-space packaging. There is a seminorm and a predicate on functions, not a normed space on equivalence classes: no quotient, no completeness, and the action is not promoted to a representation on a carrier. Nor is either operator's boundedness stated as an operator norm; what is proved are the two inputs such a statement is assembled from. The composite still needs a nonvanishing constant, and that is where the negative-order condition on the synthesis feature — the growth index, and where the rectified linear unit would have to be checked — remains open.

Lemma4.5.13

Statement uses 3

Statement dependency previews

Preview

Theorem 4.5.4

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

used by 1✓L∃∀N

The space as a Hilbert space, and the action on it. The carrier is k+1 copies of parameter L^2 in the \ell^1 norm, so the norm of the tuple of derivatives is the seminorm; the space is the closure of the span of the tuples of the smooth members, hence complete; and the action is the diagonal of the parameter representation, hence isometric, and it carries the space.

Lean code for Lemma4.5.13●34 declarations

abbrevdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

abbrev LeanRidgelet.QuadraticSobolevCarrier.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) : Type u_1

abbrev LeanRidgelet.QuadraticSobolevCarrier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) : Type u_1

Implementation after :=

:=
  PiLp 1 fun _ : Fin (k + 1) => Lp ℂ 2 (quadraticRelativeMeasure lam)

**The carrier of `Γ^k`.**  Order `k` means `k + 1` derivatives, so the carrier is `k + 1` copies
of parameter `L²`, taken in the `ℓ¹` norm.  The choice of `ℓ¹` is what makes
`LeanRidgelet.norm_quadraticSobolevJet` an equality rather than a two-sided estimate: the order-`k`
Sobolev seminorm is a plain sum of `L²` norms, and the `ℓ¹` norm of a tuple is a plain sum too.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.norm_quadraticSobolevCarrier.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) (x : LeanRidgelet.QuadraticSobolevCarrier lam k) :
  ‖x‖ = ∑ j, ‖x.ofLp j‖

theorem LeanRidgelet.norm_quadraticSobolevCarrier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ)
  (x :
    LeanRidgelet.QuadraticSobolevCarrier
      lam k) :
  ‖x‖ = ∑ j, ‖x.ofLp j‖

The norm on the carrier is the sum of the norms of the components.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.MemQuadraticSobolev.memLp_quadraticConstIteratedDeriv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  {lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)}
  {k j : ℕ} {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (h : LeanRidgelet.MemQuadraticSobolev lam k T) (hj : j ≤ k) :
  MeasureTheory.MemLp (LeanRidgelet.quadraticConstIteratedDeriv j T) 2
    (LeanRidgelet.quadraticRelativeMeasure lam)

theorem LeanRidgelet.MemQuadraticSobolev.memLp_quadraticConstIteratedDeriv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E)}
  {k j : ℕ}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (h :
    LeanRidgelet.MemQuadraticSobolev lam k
      T)
  (hj : j ≤ k) :
  MeasureTheory.MemLp
    (LeanRidgelet.quadraticConstIteratedDeriv
      j T)
    2
    (LeanRidgelet.quadraticRelativeMeasure
      lam)

Every derivative up to order `k` of a member of the space is an `L²` function for the parameter
measure: it is measurable by the membership predicate, and its `L²` norm is one term of a finite
sum.

defdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

def LeanRidgelet.quadraticSobolevJet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (h : LeanRidgelet.MemQuadraticSobolev lam k T) :
  LeanRidgelet.QuadraticSobolevCarrier lam k

def LeanRidgelet.quadraticSobolevJet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (h :
    LeanRidgelet.MemQuadraticSobolev lam k
      T) :
  LeanRidgelet.QuadraticSobolevCarrier lam
    k

Implementation after :=

:=
  WithLp.toLp 1 fun j : Fin (k + 1) ↦
    (h.memLp_quadraticConstIteratedDeriv (Nat.lt_succ_iff.1 j.isLt)).toLp _

**The tuple of derivatives.**  A member of the space is sent to the tuple of the `L²` classes of
its derivatives in the constant coefficient, up to order `k`.  This is the map whose image spans the
space.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticSobolevJet_component.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (h : LeanRidgelet.MemQuadraticSobolev lam k T) (j : Fin (k + 1)) :
  (LeanRidgelet.quadraticSobolevJet lam k h).ofLp j =
    MeasureTheory.MemLp.toLp
      (LeanRidgelet.quadraticConstIteratedDeriv (↑j) T) ⋯

theorem LeanRidgelet.quadraticSobolevJet_component.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (h :
    LeanRidgelet.MemQuadraticSobolev lam k
      T)
  (j : Fin (k + 1)) :
  (LeanRidgelet.quadraticSobolevJet lam k
          h).ofLp
      j =
    MeasureTheory.MemLp.toLp
      (LeanRidgelet.quadraticConstIteratedDeriv
        (↑j) T)
      ⋯

The components of the tuple are the `L²` classes of the derivatives.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.norm_quadraticSobolevJet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (h : LeanRidgelet.MemQuadraticSobolev lam k T) :
  ‖LeanRidgelet.quadraticSobolevJet lam k h‖ =
    (LeanRidgelet.quadraticSobolevSeminorm lam k T).toReal

theorem LeanRidgelet.norm_quadraticSobolevJet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (h :
    LeanRidgelet.MemQuadraticSobolev lam k
      T) :
  ‖LeanRidgelet.quadraticSobolevJet lam k
        h‖ =
    (LeanRidgelet.quadraticSobolevSeminorm
        lam k T).toReal

**The norm of the tuple is the seminorm.**  Not up to a constant and not up to equivalence of
norms: the `ℓ¹` norm of the tuple of derivatives is the order-`k` Sobolev seminorm of the
coefficient function.  This is the whole reason the carrier is taken in the `ℓ¹` norm.

defdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

def LeanRidgelet.quadraticSobolevJets.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) : Set (LeanRidgelet.QuadraticSobolevCarrier lam k)

def LeanRidgelet.quadraticSobolevJets.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) :
  Set
    (LeanRidgelet.QuadraticSobolevCarrier
      lam k)

Implementation after :=

:=
  {x | ∃ T : QuadraticParameter E → ℂ, ∃ h : MemQuadraticSobolev lam k T,
    ContDiff ℝ k T ∧ quadraticSobolevJet lam k h = x}

The tuples of the smooth members of the space.  Smoothness is what makes the derivative of a
sum the sum of the derivatives, so it is what makes the span below meaningful; and it is preserved
by the action, so the space is invariant.

defdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

def LeanRidgelet.quadraticSobolevSpace.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) : Submodule ℂ (LeanRidgelet.QuadraticSobolevCarrier lam k)

def LeanRidgelet.quadraticSobolevSpace.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) :
  Submodule ℂ
    (LeanRidgelet.QuadraticSobolevCarrier
      lam k)

Implementation after :=

:=
  (Submodule.span ℂ (quadraticSobolevJets lam k)).topologicalClosure

**The intermediate coefficient space `Γ^k`.**  The closure of the span of the tuples of the
smooth coefficient functions of finite order-`k` seminorm.  This is the standard construction of a
Sobolev space as a completion, and it is the construction the carrier forces: the derivative of an
almost-everywhere equivalence class is not defined, so `Γ^k` cannot be cut out of `L²` by a
pointwise condition.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.isClosed_quadraticSobolevSpace.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) : IsClosed ↑(LeanRidgelet.quadraticSobolevSpace lam k)

theorem LeanRidgelet.isClosed_quadraticSobolevSpace.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) :
  IsClosed
    ↑(LeanRidgelet.quadraticSobolevSpace
        lam k)

The space is closed in the carrier.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticSobolevJet_mem_quadraticSobolevSpace.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (k : ℕ) {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (h : LeanRidgelet.MemQuadraticSobolev lam k T)
  (hT : ContDiff ℝ (↑k) T) :
  LeanRidgelet.quadraticSobolevJet lam k h ∈
    LeanRidgelet.quadraticSobolevSpace lam k

theorem LeanRidgelet.quadraticSobolevJet_mem_quadraticSobolevSpace.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (k : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (h :
    LeanRidgelet.MemQuadraticSobolev lam k
      T)
  (hT : ContDiff ℝ (↑k) T) :
  LeanRidgelet.quadraticSobolevJet lam k
      h ∈
    LeanRidgelet.quadraticSobolevSpace lam
      k

The tuple of a smooth member lies in the space.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.contDiff_comp_quadraticParameterSMul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {m : WithTop ℕ∞} {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ m T) (g : E ≃ᵃ[ℝ] E) : ContDiff ℝ m fun η ↦ T (g • η)

theorem LeanRidgelet.contDiff_comp_quadraticParameterSMul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {m : WithTop ℕ∞}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ m T) (g : E ≃ᵃ[ℝ] E) :
  ContDiff ℝ m fun η ↦ T (g • η)

The parameter action is a linear automorphism of a finite-dimensional space, hence smooth, so a
smooth coefficient function stays smooth when pulled back along it.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.contDiff_quadraticConstSlice.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {m : WithTop ℕ∞} {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ m T) (p : LeanRidgelet.QuadraticSymmetric E × E) :
  ContDiff ℝ m (LeanRidgelet.quadraticConstSlice T p)

theorem LeanRidgelet.contDiff_quadraticConstSlice.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {m : WithTop ℕ∞}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ m T)
  (p :
    LeanRidgelet.QuadraticSymmetric E ×
      E) :
  ContDiff ℝ m
    (LeanRidgelet.quadraticConstSlice T p)

A slice of a smooth coefficient function in the constant coefficient is smooth: the slice is the
composition with an affine map of the line.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticConstIteratedDeriv_const_smul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {j k : ℕ} (hjk : j ≤ k) {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑k) T) (c : ℂ) :
  (LeanRidgelet.quadraticConstIteratedDeriv j fun η ↦ c • T η) = fun ξ ↦
    c • LeanRidgelet.quadraticConstIteratedDeriv j T ξ

theorem LeanRidgelet.quadraticConstIteratedDeriv_const_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {j k : ℕ}
  (hjk : j ≤ k)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑k) T) (c : ℂ) :
  (LeanRidgelet.quadraticConstIteratedDeriv
      j fun η ↦ c • T η) =
    fun ξ ↦
    c •
      LeanRidgelet.quadraticConstIteratedDeriv
        j T ξ

The derivative in the constant coefficient commutes with a constant multiple.  Constants are the
only multipliers this is needed for, and it is where smoothness of the coefficient function is
spent.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.radonNikodymWeight_quadraticRelativeParameterJacobian_const.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ η : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.radonNikodymWeight
      LeanRidgelet.quadraticRelativeParameterJacobian g ξ =
    LeanRidgelet.radonNikodymWeight
      LeanRidgelet.quadraticRelativeParameterJacobian g η

theorem LeanRidgelet.radonNikodymWeight_quadraticRelativeParameterJacobian_const.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (ξ η :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.radonNikodymWeight
      LeanRidgelet.quadraticRelativeParameterJacobian
      g ξ =
    LeanRidgelet.radonNikodymWeight
      LeanRidgelet.quadraticRelativeParameterJacobian
      g η

The Radon--Nikodym weight of the relatively invariant parameter measure does not depend on the
parameter: it is the reciprocal square root of the determinant of the linear part.  This is what
lets the weight pass through a derivative in the constant coefficient.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticConstIteratedDeriv_quasiRegularAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {j k : ℕ} (hjk : j ≤ k)
  {T : LeanRidgelet.QuadraticParameter E → ℂ} (hT : ContDiff ℝ (↑k) T)
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.quadraticConstIteratedDeriv j
      (LeanRidgelet.quasiRegularAction
        (LeanRidgelet.radonNikodymWeight
          LeanRidgelet.quadraticRelativeParameterJacobian)
        g T) =
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g (LeanRidgelet.quadraticConstIteratedDeriv j T)

theorem LeanRidgelet.quadraticConstIteratedDeriv_quasiRegularAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {j k : ℕ}
  (hjk : j ≤ k)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑k) T)
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.quadraticConstIteratedDeriv
      j
      (LeanRidgelet.quasiRegularAction
        (LeanRidgelet.radonNikodymWeight
          LeanRidgelet.quadraticRelativeParameterJacobian)
        g T) =
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g
      (LeanRidgelet.quadraticConstIteratedDeriv
        j T)

**The derivative in the constant coefficient commutes with the parameter representation.**  For
a smooth coefficient function, differentiating the corrected pullback in the constant coefficient is
the corrected pullback of the derivative.  Two facts combine: the correction is a constant in the
parameter, and the action is a shear in the constant coefficient, so the pullback commutes with the
derivative on the nose.  This is what makes the tuple map of this file equivariant.

defdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

def LeanRidgelet.quadraticRelativeParameterLpIsometry.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) :
  ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure lam)) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2 (LeanRidgelet.quadraticRelativeMeasure lam))

def LeanRidgelet.quadraticRelativeParameterLpIsometry.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E) :
  ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure
          lam)) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure
          lam))

Implementation after :=

:=
  Unitary.linearIsometryEquiv (quadraticRelativeParameterLpUnitaryRepresentation lam g)

The parameter representation at one group element, as an isometric equivalence of parameter
`L²`.  Only the unitarity of the representation is used.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticRelativeParameterLpIsometry_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure lam))) :
  (LeanRidgelet.quadraticRelativeParameterLpIsometry lam g) f =
    ↑((LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
            lam)
          g)
      f

theorem LeanRidgelet.quadraticRelativeParameterLpIsometry_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.quadraticRelativeMeasure
          lam))) :
  (LeanRidgelet.quadraticRelativeParameterLpIsometry
        lam g)
      f =
    ↑((LeanRidgelet.quadraticRelativeParameterLpUnitaryRepresentation
            lam)
          g)
      f

The isometric equivalence is the representation's operator.

defdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

def LeanRidgelet.quadraticSobolevCarrierAction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.QuadraticSobolevCarrier lam k ≃ₗᵢ[ℂ]
    LeanRidgelet.QuadraticSobolevCarrier lam k

def LeanRidgelet.quadraticSobolevCarrierAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.QuadraticSobolevCarrier lam
      k ≃ₗᵢ[ℂ]
    LeanRidgelet.QuadraticSobolevCarrier
      lam k

Implementation after :=

:=
  LinearIsometryEquiv.piLpCongrRight 1 fun _ ↦ quadraticRelativeParameterLpIsometry lam g

**The action on the carrier.**  The diagonal of the parameter representation: it acts on each
component of the tuple by the same operator.  Being the diagonal of a unitary, it is an isometric
equivalence of the `ℓ¹` product.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticSobolevCarrierAction_component.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) (g : E ≃ᵃ[ℝ] E)
  (x : LeanRidgelet.QuadraticSobolevCarrier lam k) (j : Fin (k + 1)) :
  ((LeanRidgelet.quadraticSobolevCarrierAction lam k g) x).ofLp j =
    (LeanRidgelet.quadraticRelativeParameterLpIsometry lam g) (x.ofLp j)

theorem LeanRidgelet.quadraticSobolevCarrierAction_component.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (g : E ≃ᵃ[ℝ] E)
  (x :
    LeanRidgelet.QuadraticSobolevCarrier
      lam k)
  (j : Fin (k + 1)) :
  ((LeanRidgelet.quadraticSobolevCarrierAction
            lam k g)
          x).ofLp
      j =
    (LeanRidgelet.quadraticRelativeParameterLpIsometry
        lam g)
      (x.ofLp j)

The action acts componentwise.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.norm_quadraticSobolevCarrierAction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) (g : E ≃ᵃ[ℝ] E)
  (x : LeanRidgelet.QuadraticSobolevCarrier lam k) :
  ‖(LeanRidgelet.quadraticSobolevCarrierAction lam k g) x‖ = ‖x‖

theorem LeanRidgelet.norm_quadraticSobolevCarrierAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (g : E ≃ᵃ[ℝ] E)
  (x :
    LeanRidgelet.QuadraticSobolevCarrier
      lam k) :
  ‖(LeanRidgelet.quadraticSobolevCarrierAction
          lam k g)
        x‖ =
    ‖x‖

The action is isometric, so it is bounded with norm one on the carrier.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticSobolevCarrierAction_one.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (x : LeanRidgelet.QuadraticSobolevCarrier lam k) :
  (LeanRidgelet.quadraticSobolevCarrierAction lam k 1) x = x

theorem LeanRidgelet.quadraticSobolevCarrierAction_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (x :
    LeanRidgelet.QuadraticSobolevCarrier
      lam k) :
  (LeanRidgelet.quadraticSobolevCarrierAction
        lam k 1)
      x =
    x

The identity acts trivially.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticSobolevCarrierAction_mul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) (g h : E ≃ᵃ[ℝ] E)
  (x : LeanRidgelet.QuadraticSobolevCarrier lam k) :
  (LeanRidgelet.quadraticSobolevCarrierAction lam k (g * h)) x =
    (LeanRidgelet.quadraticSobolevCarrierAction lam k g)
      ((LeanRidgelet.quadraticSobolevCarrierAction lam k h) x)

theorem LeanRidgelet.quadraticSobolevCarrierAction_mul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (g h : E ≃ᵃ[ℝ] E)
  (x :
    LeanRidgelet.QuadraticSobolevCarrier
      lam k) :
  (LeanRidgelet.quadraticSobolevCarrierAction
        lam k (g * h))
      x =
    (LeanRidgelet.quadraticSobolevCarrierAction
        lam k g)
      ((LeanRidgelet.quadraticSobolevCarrierAction
          lam k h)
        x)

The action is multiplicative, so it is a representation of the affine group on the carrier.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.contDiff_quasiRegularAction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {k : ℕ} {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑k) T) (g : E ≃ᵃ[ℝ] E) :
  ContDiff ℝ (↑k)
    (LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g T)

theorem LeanRidgelet.contDiff_quasiRegularAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {k : ℕ}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hT : ContDiff ℝ (↑k) T)
  (g : E ≃ᵃ[ℝ] E) :
  ContDiff ℝ (↑k)
    (LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g T)

The corrected pullback of a smooth coefficient function is smooth.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.coeFn_quadraticRelativeParameterLpIsometry_toLp.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  {u : LeanRidgelet.QuadraticParameter E → ℂ}
  (hu :
    MeasureTheory.MemLp u 2
      (LeanRidgelet.quadraticRelativeMeasure lam)) :
  ↑↑((LeanRidgelet.quadraticRelativeParameterLpIsometry lam g)
          (MeasureTheory.MemLp.toLp u
            hu)) =ᵐ[LeanRidgelet.quadraticRelativeMeasure lam]
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g u

theorem LeanRidgelet.coeFn_quadraticRelativeParameterLpIsometry_toLp.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (g : E ≃ᵃ[ℝ] E)
  {u :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hu :
    MeasureTheory.MemLp u 2
      (LeanRidgelet.quadraticRelativeMeasure
        lam)) :
  ↑↑((LeanRidgelet.quadraticRelativeParameterLpIsometry
            lam g)
          (MeasureTheory.MemLp.toLp u
            hu)) =ᵐ[LeanRidgelet.quadraticRelativeMeasure
      lam]
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g u

The image of an `L²` class under the parameter representation is represented by the corrected
pullback of any representative.  This is the bridge between the pointwise identity
`LeanRidgelet.quadraticConstIteratedDeriv_quasiRegularAction` and the `L²` classes the carrier is
built from; the transport of an almost-everywhere equality along the action is what
`LeanRidgelet.quadraticRelative_quasiRegularAction_congr_ae` supplies.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticSobolevSeminorm_quasiRegularAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {k : ℕ}
  {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (h : LeanRidgelet.MemQuadraticSobolev lam k T)
  (hT : ContDiff ℝ (↑k) T) (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.quadraticSobolevSeminorm lam k
      (LeanRidgelet.quasiRegularAction
        (LeanRidgelet.radonNikodymWeight
          LeanRidgelet.quadraticRelativeParameterJacobian)
        g T) =
    LeanRidgelet.quadraticSobolevSeminorm lam k T

theorem LeanRidgelet.quadraticSobolevSeminorm_quasiRegularAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {k : ℕ}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (h :
    LeanRidgelet.MemQuadraticSobolev lam k
      T)
  (hT : ContDiff ℝ (↑k) T)
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.quadraticSobolevSeminorm
      lam k
      (LeanRidgelet.quasiRegularAction
        (LeanRidgelet.radonNikodymWeight
          LeanRidgelet.quadraticRelativeParameterJacobian)
        g T) =
    LeanRidgelet.quadraticSobolevSeminorm
      lam k T

**The action is isometric on the seminorm.**  The corrected pullback of a smooth coefficient
function has the *same* order-`k` Sobolev seminorm.  Contrast
`LeanRidgelet.quadraticSobolevSeminorm_comp_smul`, where the plain pull-back scales the seminorm by
the square root of the determinant of the linear part: the Radon--Nikodym weight of the
representation is exactly that factor, and it cancels.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.memQuadraticSobolev_quasiRegularAction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {k : ℕ}
  {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (h : LeanRidgelet.MemQuadraticSobolev lam k T)
  (hT : ContDiff ℝ (↑k) T) (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.MemQuadraticSobolev lam k
    (LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g T)

theorem LeanRidgelet.memQuadraticSobolev_quasiRegularAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {k : ℕ}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (h :
    LeanRidgelet.MemQuadraticSobolev lam k
      T)
  (hT : ContDiff ℝ (↑k) T)
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.MemQuadraticSobolev lam k
    (LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.quadraticRelativeParameterJacobian)
      g T)

**The action carries the space.**  The corrected pullback of a smooth member is a member.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticSobolevCarrierAction_quadraticSobolevJet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {k : ℕ}
  {T : LeanRidgelet.QuadraticParameter E → ℂ}
  (h : LeanRidgelet.MemQuadraticSobolev lam k T)
  (hT : ContDiff ℝ (↑k) T) (g : E ≃ᵃ[ℝ] E) :
  (LeanRidgelet.quadraticSobolevCarrierAction lam k g)
      (LeanRidgelet.quadraticSobolevJet lam k h) =
    LeanRidgelet.quadraticSobolevJet lam k ⋯

theorem LeanRidgelet.quadraticSobolevCarrierAction_quadraticSobolevJet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {k : ℕ}
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (h :
    LeanRidgelet.MemQuadraticSobolev lam k
      T)
  (hT : ContDiff ℝ (↑k) T)
  (g : E ≃ᵃ[ℝ] E) :
  (LeanRidgelet.quadraticSobolevCarrierAction
        lam k g)
      (LeanRidgelet.quadraticSobolevJet
        lam k h) =
    LeanRidgelet.quadraticSobolevJet lam k
      ⋯

**The tuple map is equivariant.**  The action on the carrier applied to the tuple of a smooth
member is the tuple of the corrected pullback of that member.  This is what makes the space
invariant.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticSobolevCarrierAction_mem_quadraticSobolevSpace.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) (g : E ≃ᵃ[ℝ] E)
  {x : LeanRidgelet.QuadraticSobolevCarrier lam k}
  (hx : x ∈ LeanRidgelet.quadraticSobolevSpace lam k) :
  (LeanRidgelet.quadraticSobolevCarrierAction lam k g) x ∈
    LeanRidgelet.quadraticSobolevSpace lam k

theorem LeanRidgelet.quadraticSobolevCarrierAction_mem_quadraticSobolevSpace.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (g : E ≃ᵃ[ℝ] E)
  {x :
    LeanRidgelet.QuadraticSobolevCarrier
      lam k}
  (hx :
    x ∈
      LeanRidgelet.quadraticSobolevSpace
        lam k) :
  (LeanRidgelet.quadraticSobolevCarrierAction
        lam k g)
      x ∈
    LeanRidgelet.quadraticSobolevSpace lam
      k

**The space is invariant.**  The action of the affine group on the carrier maps `Γ^k` into
itself, so the representation restricts to it.  The tuples of smooth members are carried among
themselves by the previous theorem, and a continuous linear map that carries a spanning set into a
closed subspace carries the closure of its span into it.

defdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

def LeanRidgelet.quadraticSobolevSpaceAction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) (g : E ≃ᵃ[ℝ] E) :
  ↥(LeanRidgelet.quadraticSobolevSpace lam k) →L[ℂ]
    ↥(LeanRidgelet.quadraticSobolevSpace lam k)

def LeanRidgelet.quadraticSobolevSpaceAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (g : E ≃ᵃ[ℝ] E) :
  ↥(LeanRidgelet.quadraticSobolevSpace lam
        k) →L[ℂ]
    ↥(LeanRidgelet.quadraticSobolevSpace
        lam k)

Implementation after :=

:=
  (((quadraticSobolevCarrierAction lam k g).toLinearIsometry.toContinuousLinearMap).comp
      (quadraticSobolevSpace lam k).subtypeL).codRestrict (quadraticSobolevSpace lam k)
    fun x ↦ quadraticSobolevCarrierAction_mem_quadraticSobolevSpace lam k g x.2

The action restricted to the space, as a bounded operator on it.  The restriction is legitimate
because the space is invariant.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticSobolevSpaceAction_coe.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) (g : E ≃ᵃ[ℝ] E)
  (x : ↥(LeanRidgelet.quadraticSobolevSpace lam k)) :
  ↑((LeanRidgelet.quadraticSobolevSpaceAction lam k g) x) =
    (LeanRidgelet.quadraticSobolevCarrierAction lam k g) ↑x

theorem LeanRidgelet.quadraticSobolevSpaceAction_coe.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (g : E ≃ᵃ[ℝ] E)
  (x :
    ↥(LeanRidgelet.quadraticSobolevSpace
        lam k)) :
  ↑((LeanRidgelet.quadraticSobolevSpaceAction
          lam k g)
        x) =
    (LeanRidgelet.quadraticSobolevCarrierAction
        lam k g)
      ↑x

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.norm_quadraticSobolevSpaceAction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) (g : E ≃ᵃ[ℝ] E)
  (x : ↥(LeanRidgelet.quadraticSobolevSpace lam k)) :
  ‖(LeanRidgelet.quadraticSobolevSpaceAction lam k g) x‖ = ‖x‖

theorem LeanRidgelet.norm_quadraticSobolevSpaceAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (g : E ≃ᵃ[ℝ] E)
  (x :
    ↥(LeanRidgelet.quadraticSobolevSpace
        lam k)) :
  ‖(LeanRidgelet.quadraticSobolevSpaceAction
          lam k g)
        x‖ =
    ‖x‖

The restricted action is isometric, being the restriction of an isometry.

defdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

def LeanRidgelet.quadraticSobolevSpaceActionMonoidHom.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) :
  (E ≃ᵃ[ℝ] E) →*
    ↥(LeanRidgelet.quadraticSobolevSpace lam k) →L[ℂ]
      ↥(LeanRidgelet.quadraticSobolevSpace lam k)

def LeanRidgelet.quadraticSobolevSpaceActionMonoidHom.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) :
  (E ≃ᵃ[ℝ] E) →*
    ↥(LeanRidgelet.quadraticSobolevSpace
          lam k) →L[ℂ]
      ↥(LeanRidgelet.quadraticSobolevSpace
          lam k)

Implementation after :=

:= quadraticSobolevSpaceAction lam k
  map_one' := ContinuousLinearMap.ext fun x ↦ Subtype.ext
    (quadraticSobolevCarrierAction_one lam k (x : QuadraticSobolevCarrier lam k))
  map_mul' g h := ContinuousLinearMap.ext fun x ↦ Subtype.ext
    (quadraticSobolevCarrierAction_mul lam k g h (x : QuadraticSobolevCarrier lam k))

The restricted action is multiplicative and unital, as a monoid homomorphism into the bounded
operators on the space.

defdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

def LeanRidgelet.quadraticSobolevContRepresentation.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) :
  ContRepresentation ℂ (E ≃ᵃ[ℝ] E)
    ↥(LeanRidgelet.quadraticSobolevSpace lam k)

def LeanRidgelet.quadraticSobolevContRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) :
  ContRepresentation ℂ (E ≃ᵃ[ℝ] E)
    ↥(LeanRidgelet.quadraticSobolevSpace
        lam k)

Implementation after :=

:=
  .ofMonoidHom (quadraticSobolevSpaceActionMonoidHom lam k)

**The parameter representation on the intermediate coefficient space.**  The affine group acts
on `Γ^k` by the restriction of the diagonal of the parameter representation, isometrically.  This is
the packaging the boundedness route needs: with it, a machine out of `Γ^k` and a ridgelet transform
into it are intertwiners of representations, so the reconstruction argument applies through the
intermediate space rather than through parameter `L²`.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevCarrier.lean

complete

theorem LeanRidgelet.quadraticSobolevContRepresentation_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) (g : E ≃ᵃ[ℝ] E) :
  (LeanRidgelet.quadraticSobolevContRepresentation lam k) g =
    LeanRidgelet.quadraticSobolevSpaceAction lam k g

theorem LeanRidgelet.quadraticSobolevContRepresentation_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (g : E ≃ᵃ[ℝ] E) :
  (LeanRidgelet.quadraticSobolevContRepresentation
        lam k)
      g =
    LeanRidgelet.quadraticSobolevSpaceAction
      lam k g

Three choices make this short. The carrier is a finite product of copies of parameter L^2 rather than anything new, because the seminorm is a sum of L^2 norms; and the product is taken in the \ell^1 norm, because then the norm of the tuple of derivatives is the seminorm on the nose — no equivalence of norms and no constants anywhere downstream. Completeness is then free: a finite product of complete spaces is complete, and the space is a closed subspace of it.

The space is the closure of the span of the tuples of the smooth coefficient functions of finite seminorm. That is forced, not chosen: the derivative of an almost-everywhere equivalence class is not defined, so a Sobolev space cannot be cut out of L^2 by a pointwise condition, and the alternative to a completion would be a theory of weak derivatives in the parameter, which this development does not have. Smoothness also makes the span meaningful, since the derivative of a sum is the sum of the derivatives only for differentiable summands.

The action is the diagonal of the parameter representation of the relatively invariant measure — no new representation is built. Two facts make the diagonal work. The representation's Radon--Nikodym weight is a constant in the parameter, the reciprocal square root of the determinant of the linear part, so it passes through a derivative; and the action is a shear in the constant coefficient, so the pullback commutes with the derivative. Together they say the tuple map is equivariant, and the diagonal of a unitary is an isometry of the \ell^1 product. The seminorm is therefore exactly preserved, in contrast with the plain pull-back, which scales it by the square root of that determinant: the weight is precisely the compensating factor.

Because the space is invariant, the action restricts to it, and the restriction is bundled as a representation of the affine group on the space — isometrically. That is what lets the reconstruction argument run through the intermediate space: a machine out of it and a ridgelet transform into it are intertwiners of representations, so Schur applies to their composite exactly as it does for parameter L^2. It is also the only pair shape the two bounds can produce, since the synthesis is bounded on the space and not on parameter L^2.

What this leaves for the boundedness statement is the factorization of the two Bochner integrals through the space. The analysis side has to land in it and the synthesis side has to be bounded on it; the estimates for both are stated as hypotheses in the endpoint chapter node below, and it is their promotion to operators on this carrier that the first of the two placeholders there still needs.

Lemma4.5.14

Statement uses 3

Statement dependency previews

Preview

Lemma 4.5.7

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

used by 0✓L∃∀N

The two candidates were one space. Plancherel in the constant coefficient turns the order-k Sobolev seminorm into a frequency-weighted L^2 norm of the angular Fourier transform, slice by slice and then over the base; and on the frequency side the data variable enters the synthesis pairing only as a phase.

Lean code for Lemma4.5.14●3 theorems

theoremdefined in LeanRidgelet/HA/QuadraticSobolevFourier.lean

complete

theorem LeanRidgelet.eLpNorm_pow_smul_angularFourier_quadraticConstSlice.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {T : LeanRidgelet.QuadraticParameter E → ℂ}
  {N : ℕ∞} {j : ℕ} (p : LeanRidgelet.QuadraticSymmetric E × E)
  (hT : ContDiff ℝ (↑N) T)
  (hint :
    ∀ (n : ℕ),
      ↑n ≤ N →
        MeasureTheory.Integrable
          (iteratedDeriv n (LeanRidgelet.quadraticConstSlice T p))
          MeasureTheory.volume)
  (hj : ↑j ≤ N)
  (hmem :
    MeasureTheory.MemLp
      (iteratedDeriv j (LeanRidgelet.quadraticConstSlice T p)) 2
      MeasureTheory.volume) :
  MeasureTheory.eLpNorm
      (fun ζ ↦
        |ζ| ^ j •
          LeanRidgelet.Fourier.angularFourierIntegralInner
            (LeanRidgelet.quadraticConstSlice T p) ζ)
      2 MeasureTheory.volume =
    ENNReal.ofReal √(2 * Real.pi) *
      MeasureTheory.eLpNorm
        (iteratedDeriv j (LeanRidgelet.quadraticConstSlice T p)) 2
        MeasureTheory.volume

theorem LeanRidgelet.eLpNorm_pow_smul_angularFourier_quadraticConstSlice.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  {N : ℕ∞} {j : ℕ}
  (p :
    LeanRidgelet.QuadraticSymmetric E × E)
  (hT : ContDiff ℝ (↑N) T)
  (hint :
    ∀ (n : ℕ),
      ↑n ≤ N →
        MeasureTheory.Integrable
          (iteratedDeriv n
            (LeanRidgelet.quadraticConstSlice
              T p))
          MeasureTheory.volume)
  (hj : ↑j ≤ N)
  (hmem :
    MeasureTheory.MemLp
      (iteratedDeriv j
        (LeanRidgelet.quadraticConstSlice
          T p))
      2 MeasureTheory.volume) :
  MeasureTheory.eLpNorm
      (fun ζ ↦
        |ζ| ^ j •
          LeanRidgelet.Fourier.angularFourierIntegralInner
            (LeanRidgelet.quadraticConstSlice
              T p)
            ζ)
      2 MeasureTheory.volume =
    ENNReal.ofReal √(2 * Real.pi) *
      MeasureTheory.eLpNorm
        (iteratedDeriv j
          (LeanRidgelet.quadraticConstSlice
            T p))
        2 MeasureTheory.volume

**The seminorm of a slice, on the frequency side.**  For a slice of a coefficient function in
the constant coefficient, the `L²` norm of the angular Fourier transform weighted by the `j`-th
power of the frequency is `√(2π)` times the `L²` norm of the `j`-th derivative of the slice.  This
is the one-variable weighted Sobolev identity of `LeanRidgelet.Fourier.AngularWeightedSobolev`
instantiated at a slice; the constant is the square root of the Plancherel constant of the angular
convention.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevFourier.lean

complete

theorem LeanRidgelet.quadraticBaseSobolevSeminorm_eq_angularFourier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (κ : MeasureTheory.Measure (LeanRidgelet.QuadraticSymmetric E × E))
  (k : ℕ) {T : LeanRidgelet.QuadraticParameter E → ℂ} {N : ℕ∞}
  (hT : ContDiff ℝ (↑N) T)
  (hint :
    ∀ (p : LeanRidgelet.QuadraticSymmetric E × E) (n : ℕ),
      ↑n ≤ N →
        MeasureTheory.Integrable
          (iteratedDeriv n (LeanRidgelet.quadraticConstSlice T p))
          MeasureTheory.volume)
  (hk : ↑k ≤ N)
  (hmem :
    ∀ (p : LeanRidgelet.QuadraticSymmetric E × E),
      ∀ j ≤ k,
        MeasureTheory.MemLp
          (iteratedDeriv j (LeanRidgelet.quadraticConstSlice T p)) 2
          MeasureTheory.volume) :
  ∑ j ∈ Finset.range (k + 1),
      (∫⁻ (p : LeanRidgelet.QuadraticSymmetric E × E),
          MeasureTheory.eLpNorm
              (fun ζ ↦
                |ζ| ^ j •
                  LeanRidgelet.Fourier.angularFourierIntegralInner
                    (LeanRidgelet.quadraticConstSlice T p) ζ)
              2 MeasureTheory.volume ^
            2 ∂LeanRidgelet.quadraticBaseRelativeMeasure κ) ^
        (1 / 2) =
    ENNReal.ofReal √(2 * Real.pi) *
      LeanRidgelet.quadraticBaseSobolevSeminorm κ k T

theorem LeanRidgelet.quadraticBaseSobolevSeminorm_eq_angularFourier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (κ :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticSymmetric E ×
        E))
  (k : ℕ)
  {T :
    LeanRidgelet.QuadraticParameter E → ℂ}
  {N : ℕ∞} (hT : ContDiff ℝ (↑N) T)
  (hint :
    ∀
      (p :
        LeanRidgelet.QuadraticSymmetric
            E ×
          E)
      (n : ℕ),
      ↑n ≤ N →
        MeasureTheory.Integrable
          (iteratedDeriv n
            (LeanRidgelet.quadraticConstSlice
              T p))
          MeasureTheory.volume)
  (hk : ↑k ≤ N)
  (hmem :
    ∀
      (p :
        LeanRidgelet.QuadraticSymmetric
            E ×
          E),
      ∀ j ≤ k,
        MeasureTheory.MemLp
          (iteratedDeriv j
            (LeanRidgelet.quadraticConstSlice
              T p))
          2 MeasureTheory.volume) :
  ∑ j ∈ Finset.range (k + 1),
      (∫⁻ (p :
          LeanRidgelet.QuadraticSymmetric
              E ×
            E),
          MeasureTheory.eLpNorm
              (fun ζ ↦
                |ζ| ^ j •
                  LeanRidgelet.Fourier.angularFourierIntegralInner
                    (LeanRidgelet.quadraticConstSlice
                      T p)
                    ζ)
              2 MeasureTheory.volume ^
            2 ∂LeanRidgelet.quadraticBaseRelativeMeasure
            κ) ^
        (1 / 2) =
    ENNReal.ofReal √(2 * Real.pi) *
      LeanRidgelet.quadraticBaseSobolevSeminorm
        κ k T

**The intermediate space is the weighted space of the other route.**  The base-level order-`k`
Sobolev seminorm in the constant coefficient is, up to the single constant `√(2π)`, the sum over
`j ≤ k` of the base integrals of the squared frequency-weighted angular Fourier transforms of the
slices.  So the space designed in the constant coefficient, where the action is a shear and the
invariance is visible, coincides with the frequency-weighted space in which the synthesis pairing is
visible.  The two candidates for the intermediate space were one space.

theoremdefined in LeanRidgelet/HA/QuadraticSobolevFourier.lean

complete

theorem LeanRidgelet.angularFourierIntegralInner_comp_const_add (σ : ℝ → ℂ)
  (a ζ : ℝ) :
  LeanRidgelet.Fourier.angularFourierIntegralInner (fun c ↦ σ (a + c))
      ζ =
    Complex.exp (Complex.I * ↑(a * ζ)) *
      LeanRidgelet.Fourier.angularFourierIntegralInner σ ζ

theorem LeanRidgelet.angularFourierIntegralInner_comp_const_add
  (σ : ℝ → ℂ) (a ζ : ℝ) :
  LeanRidgelet.Fourier.angularFourierIntegralInner
      (fun c ↦ σ (a + c)) ζ =
    Complex.exp (Complex.I * ↑(a * ζ)) *
      LeanRidgelet.Fourier.angularFourierIntegralInner
        σ ζ

**Translating the argument multiplies the transform by a phase.**  In the constant coefficient
the data variable enters the synthesis pairing only by translating the activation, so on the
frequency side it enters only through this phase.  Nothing is assumed: for a function whose defining
integral diverges both sides are zero, which is exactly the case of an activation of polynomial
growth, and the reason the synthesis pairing has to be read distributionally.

The intermediate space was designed in the constant coefficient rather than in its dual, because the action is a shear there and the Sobolev structure is therefore carried along — transporting to the dual side was the route not taken. This node shows the two routes describe the same space, so the choice was a choice of which side to make the invariance visible on. The invariance is visible in the constant coefficient, where the action is a shear. The synthesis bound is visible on the frequency side, where the activation appears through its own Fourier transform.

That is where the activation enters, and the shape is now explicit. For a frozen base parameter the synthesis integral is a pairing \int \gamma(c)\,\sigma(a+c)\,dc with a the part of the scalar argument free of c, so the data variable enters only by translating the activation — and a translation is a phase on the frequency side. The pairing becomes an integral of the transform of the coefficient function against the transform of the activation, modulated by that phase, and the frequency weight |\zeta|^k the seminorm carries is what has to absorb the singularity of the activation's transform. For the rectified linear unit that transform has a second-order pole at the origin, so k\ge 2 is the natural order. That is the negative-order condition on the synthesis feature, made concrete.

It is also why the synthesis bound is not a theorem yet, and the modulation identity is what shows it: an activation of polynomial growth has no Fourier transform as a function, the defining integral not converging, so both sides of the identity degenerate to zero for the rectified linear unit. The pairing is distributional and Parseval does not literally apply. A theorem needs a duality statement against the distributional transform. What the identity settles is the shape — data variable as phase, activation as a fixed frequency profile, order k as what pairs against it — and that shape is what a duality statement would have to be written against.

Theorem4.5.15

Statement uses 3

Statement dependency previews

Preview

Theorem 4.5.2

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

used by 0✓L∃∀N

The rectified linear unit reduces to a hat function. Build a second difference into the analysis feature and the composite against an activation of polynomial growth becomes the composite against a bounded, compactly supported one — with no distributional pairing anywhere, a second difference being a finite combination of translations.

Lean code for Theorem4.5.15●35 declarations

defdefined in LeanRidgelet/HA/SecondDifference.lean

complete

def LeanRidgelet.secondDifference (h : ℝ) (g : ℝ → ℂ) : ℝ → ℂ

def LeanRidgelet.secondDifference (h : ℝ)
  (g : ℝ → ℂ) : ℝ → ℂ

Implementation after :=

:=
  fun z ↦ g (z + h) - 2 * g z + g (z - h)

**The second difference at step `h`.**  A finite combination of three translations, so it moves
across an integral by translation invariance alone.

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.secondDifference_apply (h : ℝ) (g : ℝ → ℂ) (z : ℝ) :
  LeanRidgelet.secondDifference h g z = g (z + h) - 2 * g z + g (z - h)

theorem LeanRidgelet.secondDifference_apply
  (h : ℝ) (g : ℝ → ℂ) (z : ℝ) :
  LeanRidgelet.secondDifference h g z =
    g (z + h) - 2 * g z + g (z - h)

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.secondDifference_neg (h : ℝ) (g : ℝ → ℂ) :
  LeanRidgelet.secondDifference (-h) g =
    LeanRidgelet.secondDifference h g

theorem LeanRidgelet.secondDifference_neg (h : ℝ)
  (g : ℝ → ℂ) :
  LeanRidgelet.secondDifference (-h) g =
    LeanRidgelet.secondDifference h g

The second difference is symmetric in the step.

defdefined in LeanRidgelet/HA/SecondDifference.lean

complete

def LeanRidgelet.hatFunction (h : ℝ) : ℝ → ℝ

def LeanRidgelet.hatFunction (h : ℝ) : ℝ → ℝ

Implementation after :=

:=
  fun z ↦ max 0 (h - |z|)

The hat function of width `2h` and height `h`: the second difference of the rectified linear
unit.

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.hatFunction_nonneg (h z : ℝ) :
  0 ≤ LeanRidgelet.hatFunction h z

theorem LeanRidgelet.hatFunction_nonneg
  (h z : ℝ) :
  0 ≤ LeanRidgelet.hatFunction h z

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.hatFunction_le (h z : ℝ) :
  LeanRidgelet.hatFunction h z ≤ max 0 h

theorem LeanRidgelet.hatFunction_le (h z : ℝ) :
  LeanRidgelet.hatFunction h z ≤ max 0 h

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.hatFunction_eq_zero_of_le {h z : ℝ} (hz : h ≤ |z|) :
  LeanRidgelet.hatFunction h z = 0

theorem LeanRidgelet.hatFunction_eq_zero_of_le
  {h z : ℝ} (hz : h ≤ |z|) :
  LeanRidgelet.hatFunction h z = 0

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.hatFunction_zero (h : ℝ) :
  LeanRidgelet.hatFunction h 0 = max 0 h

theorem LeanRidgelet.hatFunction_zero (h : ℝ) :
  LeanRidgelet.hatFunction h 0 = max 0 h

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.continuous_hatFunction (h : ℝ) :
  Continuous (LeanRidgelet.hatFunction h)

theorem LeanRidgelet.continuous_hatFunction
  (h : ℝ) :
  Continuous (LeanRidgelet.hatFunction h)

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.hasCompactSupport_hatFunction (h : ℝ) :
  HasCompactSupport (LeanRidgelet.hatFunction h)

theorem LeanRidgelet.hasCompactSupport_hatFunction
  (h : ℝ) :
  HasCompactSupport
    (LeanRidgelet.hatFunction h)

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.integrable_hatFunction (h : ℝ) :
  MeasureTheory.Integrable (LeanRidgelet.hatFunction h)
    MeasureTheory.volume

theorem LeanRidgelet.integrable_hatFunction
  (h : ℝ) :
  MeasureTheory.Integrable
    (LeanRidgelet.hatFunction h)
    MeasureTheory.volume

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.secondDifference_relu (h : ℝ) (hh : 0 ≤ h) (z : ℝ) :
  LeanRidgelet.secondDifference h (fun t ↦ ↑(LeanRidgelet.relu t)) z =
    ↑(LeanRidgelet.hatFunction h z)

theorem LeanRidgelet.secondDifference_relu (h : ℝ)
  (hh : 0 ≤ h) (z : ℝ) :
  LeanRidgelet.secondDifference h
      (fun t ↦ ↑(LeanRidgelet.relu t)) z =
    ↑(LeanRidgelet.hatFunction h z)

**The second difference of the rectified linear unit is the hat function.**  The unit is linear
on each side of the origin, so the difference vanishes outside `[-h, h]`; inside, the three pieces
combine to `h - |z|`.

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.integral_hatFunction_pos {h : ℝ} (hh : 0 < h) :
  0 < ∫ (z : ℝ), LeanRidgelet.hatFunction h z

theorem LeanRidgelet.integral_hatFunction_pos
  {h : ℝ} (hh : 0 < h) :
  0 <
    ∫ (z : ℝ),
      LeanRidgelet.hatFunction h z

**The hat function has positive integral.**  It is continuous, nonnegative, and positive at the
origin, so its support has positive measure.  The value is `h²`; only positivity is needed below,
and it is what makes the reduced activation's admissibility constant nonzero.

defdefined in LeanRidgelet/HA/SecondDifference.lean

complete

def LeanRidgelet.reluComplex : ℝ → ℂ

def LeanRidgelet.reluComplex : ℝ → ℂ

Implementation after :=

:= fun z ↦ ((relu z : ℝ) : ℂ)

The rectified linear unit as a complex-valued activation.

defdefined in LeanRidgelet/HA/SecondDifference.lean

complete

def LeanRidgelet.hatComplex (h : ℝ) : ℝ → ℂ

def LeanRidgelet.hatComplex (h : ℝ) : ℝ → ℂ

Implementation after :=

:= fun z ↦ ((hatFunction h z : ℝ) : ℂ)

The hat function as a complex-valued activation.

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.continuous_hatComplex (h : ℝ) :
  Continuous (LeanRidgelet.hatComplex h)

theorem LeanRidgelet.continuous_hatComplex
  (h : ℝ) :
  Continuous (LeanRidgelet.hatComplex h)

The complex hat function is continuous.

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.norm_hatComplex_le (h z : ℝ) :
  ‖LeanRidgelet.hatComplex h z‖ ≤ max 0 h

theorem LeanRidgelet.norm_hatComplex_le
  (h z : ℝ) :
  ‖LeanRidgelet.hatComplex h z‖ ≤ max 0 h

The complex hat function is bounded by the height of the hat.

theoremdefined in LeanRidgelet/HA/SecondDifference.lean

complete

theorem LeanRidgelet.secondDifference_reluComplex {h : ℝ} (hh : 0 ≤ h) :
  LeanRidgelet.secondDifference h LeanRidgelet.reluComplex =
    LeanRidgelet.hatComplex h

theorem LeanRidgelet.secondDifference_reluComplex
  {h : ℝ} (hh : 0 ≤ h) :
  LeanRidgelet.secondDifference h
      LeanRidgelet.reluComplex =
    LeanRidgelet.hatComplex h

**The second difference of the rectified linear unit is the hat function**, as complex-valued
activations.

defdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

def LeanRidgelet.quadraticConstSecondDifference.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (h : ℝ) (T : LeanRidgelet.QuadraticParameter E → ℂ) :
  LeanRidgelet.QuadraticParameter E → ℂ

def LeanRidgelet.quadraticConstSecondDifference.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ)
  (T :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  LeanRidgelet.QuadraticParameter E → ℂ

Implementation after :=

:=
  fun ξ ↦ T (ξ.1, ξ.2.1, ξ.2.2 + h) - 2 * T ξ + T (ξ.1, ξ.2.1, ξ.2.2 - h)

The second difference in the constant coefficient of a coefficient function on the parameter
space.

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.quadraticConstSecondDifference_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (h : ℝ) (T : LeanRidgelet.QuadraticParameter E → ℂ)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticConstSecondDifference h T ξ =
    T (ξ.1, ξ.2.1, ξ.2.2 + h) - 2 * T ξ + T (ξ.1, ξ.2.1, ξ.2.2 - h)

theorem LeanRidgelet.quadraticConstSecondDifference_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ)
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticConstSecondDifference
      h T ξ =
    T (ξ.1, ξ.2.1, ξ.2.2 + h) - 2 * T ξ +
      T (ξ.1, ξ.2.1, ξ.2.2 - h)

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.quadraticConstSecondDifference_eq_slice.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ)
  (T : LeanRidgelet.QuadraticParameter E → ℂ)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticConstSecondDifference h T ξ =
    LeanRidgelet.secondDifference h
      (LeanRidgelet.quadraticConstSlice T (ξ.1, ξ.2.1)) ξ.2.2

theorem LeanRidgelet.quadraticConstSecondDifference_eq_slice.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ)
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticConstSecondDifference
      h T ξ =
    LeanRidgelet.secondDifference h
      (LeanRidgelet.quadraticConstSlice T
        (ξ.1, ξ.2.1))
      ξ.2.2

The second difference in the constant coefficient is the second difference of the slice.

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.quadraticConstSecondDifference_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ)
  (T : LeanRidgelet.QuadraticParameter E → ℂ) (g : E ≃ᵃ[ℝ] E)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticConstSecondDifference h (fun η ↦ T (g • η)) ξ =
    LeanRidgelet.quadraticConstSecondDifference h T (g • ξ)

theorem LeanRidgelet.quadraticConstSecondDifference_comp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ)
  (T :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (g : E ≃ᵃ[ℝ] E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticConstSecondDifference
      h (fun η ↦ T (g • η)) ξ =
    LeanRidgelet.quadraticConstSecondDifference
      h T (g • ξ)

**The difference commutes with the parameter action.**  The action is a shear in the constant
coefficient, so it translates that coefficient by an amount depending only on the other two, and a
translation commutes with a difference on the nose.  This is the difference analogue of
`LeanRidgelet.quadraticConstIteratedDeriv_comp_smul`, and it needs no differentiability.

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.quadraticArgument_const_shift.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (t : ℝ) (x : E) (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticArgument x (ξ.1, ξ.2.1, ξ.2.2 + t) =
    LeanRidgelet.quadraticArgument x ξ + t

theorem LeanRidgelet.quadraticArgument_const_shift.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (t : ℝ) (x : E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticArgument x
      (ξ.1, ξ.2.1, ξ.2.2 + t) =
    LeanRidgelet.quadraticArgument x ξ + t

The constant coefficient enters the scalar argument additively, so shifting it shifts the
argument.

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.bochnerRidgelet_secondDifference_feature.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (h : ℝ)
  {ψ : ℝ → ℂ} (f : E → ℂ) (ξ : LeanRidgelet.QuadraticParameter E)
  (hplus :
    MeasureTheory.Integrable
      (fun x ↦
        f x *
          (starRingEnd ℂ) (ψ (LeanRidgelet.quadraticArgument x ξ + h)))
      MeasureTheory.volume)
  (hzero :
    MeasureTheory.Integrable
      (fun x ↦
        f x * (starRingEnd ℂ) (ψ (LeanRidgelet.quadraticArgument x ξ)))
      MeasureTheory.volume)
  (hminus :
    MeasureTheory.Integrable
      (fun x ↦
        f x *
          (starRingEnd ℂ) (ψ (LeanRidgelet.quadraticArgument x ξ - h)))
      MeasureTheory.volume) :
  LeanRidgelet.bochnerRidgelet MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature
        (LeanRidgelet.secondDifference h ψ))
      f ξ =
    LeanRidgelet.quadraticConstSecondDifference h
      (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature ψ) f)
      ξ

theorem LeanRidgelet.bochnerRidgelet_secondDifference_feature.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (h : ℝ) {ψ : ℝ → ℂ} (f : E → ℂ)
  (ξ : LeanRidgelet.QuadraticParameter E)
  (hplus :
    MeasureTheory.Integrable
      (fun x ↦
        f x *
          (starRingEnd ℂ)
            (ψ
              (LeanRidgelet.quadraticArgument
                  x ξ +
                h)))
      MeasureTheory.volume)
  (hzero :
    MeasureTheory.Integrable
      (fun x ↦
        f x *
          (starRingEnd ℂ)
            (ψ
              (LeanRidgelet.quadraticArgument
                x ξ)))
      MeasureTheory.volume)
  (hminus :
    MeasureTheory.Integrable
      (fun x ↦
        f x *
          (starRingEnd ℂ)
            (ψ
              (LeanRidgelet.quadraticArgument
                  x ξ -
                h)))
      MeasureTheory.volume) :
  LeanRidgelet.bochnerRidgelet
      MeasureTheory.volume
      (LeanRidgelet.quadraticVectorFeature
        (LeanRidgelet.secondDifference h
          ψ))
      f ξ =
    LeanRidgelet.quadraticConstSecondDifference
      h
      (LeanRidgelet.bochnerRidgelet
        MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature
          ψ)
        f)
      ξ

**Differencing the analysis feature differences the transform.**  The constant coefficient
enters
the feature additively, so a difference in it acts on the data integral's integrand pointwise; the
only hypothesis is the integrability that lets one integral be split into three.

defdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

def LeanRidgelet.quadraticConstTranslate.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)] (t : ℝ) :
  LeanRidgelet.QuadraticParameter E ≃ᵐ LeanRidgelet.QuadraticParameter E

def LeanRidgelet.quadraticConstTranslate.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (t : ℝ) :
  LeanRidgelet.QuadraticParameter E ≃ᵐ
    LeanRidgelet.QuadraticParameter E

Implementation after :=

:=
  (MeasurableEquiv.refl (QuadraticSymmetric E)).prodCongr
    ((MeasurableEquiv.refl E).prodCongr (MeasurableEquiv.addRight t))

The translation of the constant coefficient, as a measurable equivalence of the parameter
space.

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.quadraticConstTranslate_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)] (t : ℝ)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  (LeanRidgelet.quadraticConstTranslate t) ξ = (ξ.1, ξ.2.1, ξ.2.2 + t)

theorem LeanRidgelet.quadraticConstTranslate_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (t : ℝ)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  (LeanRidgelet.quadraticConstTranslate t)
      ξ =
    (ξ.1, ξ.2.1, ξ.2.2 + t)

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.quadraticArgument_quadraticConstTranslate.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)] (t : ℝ) (x : E)
  (ξ : LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticArgument x
      ((LeanRidgelet.quadraticConstTranslate t) ξ) =
    LeanRidgelet.quadraticArgument x ξ + t

theorem LeanRidgelet.quadraticArgument_quadraticConstTranslate.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (t : ℝ) (x : E)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  LeanRidgelet.quadraticArgument x
      ((LeanRidgelet.quadraticConstTranslate
          t)
        ξ) =
    LeanRidgelet.quadraticArgument x ξ + t

The scalar argument of the feature at a translated parameter is the argument translated: the
constant coefficient enters additively.

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.integral_comp_quadraticConstTranslate.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  {ν : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)}
  {t : ℝ}
  (hν :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate t)) ν =
      ν)
  (F : LeanRidgelet.QuadraticParameter E → ℂ) :
  ∫ (ξ : LeanRidgelet.QuadraticParameter E),
      F ((LeanRidgelet.quadraticConstTranslate t) ξ) ∂ν =
    ∫ (ξ : LeanRidgelet.QuadraticParameter E), F ξ ∂ν

theorem LeanRidgelet.integral_comp_quadraticConstTranslate.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {ν :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E)}
  {t : ℝ}
  (hν :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate
            t))
        ν =
      ν)
  (F :
    LeanRidgelet.QuadraticParameter E →
      ℂ) :
  ∫ (ξ :
      LeanRidgelet.QuadraticParameter E),
      F
        ((LeanRidgelet.quadraticConstTranslate
            t)
          ξ) ∂ν =
    ∫ (ξ :
      LeanRidgelet.QuadraticParameter E),
      F ξ ∂ν

Integrating a translated integrand against a translation-invariant measure changes nothing.

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.integrable_comp_quadraticConstTranslate.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  {ν : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)}
  {t : ℝ}
  (hν :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate t)) ν =
      ν)
  {F : LeanRidgelet.QuadraticParameter E → ℂ}
  (hF : MeasureTheory.Integrable F ν) :
  MeasureTheory.Integrable
    (fun ξ ↦ F ((LeanRidgelet.quadraticConstTranslate t) ξ)) ν

theorem LeanRidgelet.integrable_comp_quadraticConstTranslate.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {ν :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E)}
  {t : ℝ}
  (hν :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate
            t))
        ν =
      ν)
  {F :
    LeanRidgelet.QuadraticParameter E → ℂ}
  (hF : MeasureTheory.Integrable F ν) :
  MeasureTheory.Integrable
    (fun ξ ↦
      F
        ((LeanRidgelet.quadraticConstTranslate
            t)
          ξ))
    ν

Integrability transports along the translation.

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.bochnerSynthesis_quadraticConstSecondDifference.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  {ν : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)}
  {h : ℝ}
  (hνp :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate h)) ν =
      ν)
  (hνm :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate (-h))) ν =
      ν)
  {σ : ℝ → ℂ} (γ : LeanRidgelet.QuadraticParameter E → ℂ) (x : E)
  (hplus :
    MeasureTheory.Integrable
      (fun ξ ↦ γ ξ * σ (LeanRidgelet.quadraticArgument x ξ + h)) ν)
  (hzero :
    MeasureTheory.Integrable
      (fun ξ ↦ γ ξ * σ (LeanRidgelet.quadraticArgument x ξ)) ν)
  (hminus :
    MeasureTheory.Integrable
      (fun ξ ↦ γ ξ * σ (LeanRidgelet.quadraticArgument x ξ - h)) ν) :
  LeanRidgelet.bochnerSynthesis ν
      (LeanRidgelet.quadraticVectorFeature σ)
      (LeanRidgelet.quadraticConstSecondDifference h γ) x =
    LeanRidgelet.bochnerSynthesis ν
      (LeanRidgelet.quadraticVectorFeature
        (LeanRidgelet.secondDifference h σ))
      γ x

theorem LeanRidgelet.bochnerSynthesis_quadraticConstSecondDifference.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {ν :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E)}
  {h : ℝ}
  (hνp :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate
            h))
        ν =
      ν)
  (hνm :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate
            (-h)))
        ν =
      ν)
  {σ : ℝ → ℂ}
  (γ :
    LeanRidgelet.QuadraticParameter E → ℂ)
  (x : E)
  (hplus :
    MeasureTheory.Integrable
      (fun ξ ↦
        γ ξ *
          σ
            (LeanRidgelet.quadraticArgument
                x ξ +
              h))
      ν)
  (hzero :
    MeasureTheory.Integrable
      (fun ξ ↦
        γ ξ *
          σ
            (LeanRidgelet.quadraticArgument
              x ξ))
      ν)
  (hminus :
    MeasureTheory.Integrable
      (fun ξ ↦
        γ ξ *
          σ
            (LeanRidgelet.quadraticArgument
                x ξ -
              h))
      ν) :
  LeanRidgelet.bochnerSynthesis ν
      (LeanRidgelet.quadraticVectorFeature
        σ)
      (LeanRidgelet.quadraticConstSecondDifference
        h γ)
      x =
    LeanRidgelet.bochnerSynthesis ν
      (LeanRidgelet.quadraticVectorFeature
        (LeanRidgelet.secondDifference h
          σ))
      γ x

**Summation by parts in the constant coefficient.**  For a parameter measure invariant under
translation of the constant coefficient, the synthesis integral of a differenced coefficient
function
against an activation is the synthesis integral of the coefficient function against the differenced
activation.  Three translations and three changes of variable; no integration by parts, no boundary
term, and no hypothesis on the activation.

This is what reduces an activation of polynomial growth to a bounded one: combined with
`LeanRidgelet.secondDifference_relu`, the right-hand side is a synthesis integral against the hat
function.

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.bochnerSynthesis_bochnerRidgelet_reluComplex_secondDifference.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  {ν : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)}
  {h : ℝ} (hh : 0 ≤ h)
  (hνp :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate h)) ν =
      ν)
  (hνm :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate (-h))) ν =
      ν)
  {ψ₀ : ℝ → ℂ} (f : E → ℂ) (x : E)
  (hRplus :
    ∀ (ξ : LeanRidgelet.QuadraticParameter E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ)
              (ψ₀ (LeanRidgelet.quadraticArgument y ξ + h)))
        MeasureTheory.volume)
  (hRzero :
    ∀ (ξ : LeanRidgelet.QuadraticParameter E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ) (ψ₀ (LeanRidgelet.quadraticArgument y ξ)))
        MeasureTheory.volume)
  (hRminus :
    ∀ (ξ : LeanRidgelet.QuadraticParameter E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ)
              (ψ₀ (LeanRidgelet.quadraticArgument y ξ - h)))
        MeasureTheory.volume)
  (hSplus :
    MeasureTheory.Integrable
      (fun ξ ↦
        LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature ψ₀) f ξ *
          LeanRidgelet.reluComplex
            (LeanRidgelet.quadraticArgument x ξ + h))
      ν)
  (hSzero :
    MeasureTheory.Integrable
      (fun ξ ↦
        LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature ψ₀) f ξ *
          LeanRidgelet.reluComplex (LeanRidgelet.quadraticArgument x ξ))
      ν)
  (hSminus :
    MeasureTheory.Integrable
      (fun ξ ↦
        LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature ψ₀) f ξ *
          LeanRidgelet.reluComplex
            (LeanRidgelet.quadraticArgument x ξ - h))
      ν) :
  LeanRidgelet.bochnerSynthesis ν
      (LeanRidgelet.quadraticVectorFeature LeanRidgelet.reluComplex)
      (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature
          (LeanRidgelet.secondDifference h ψ₀))
        f)
      x =
    LeanRidgelet.bochnerSynthesis ν
      (LeanRidgelet.quadraticVectorFeature (LeanRidgelet.hatComplex h))
      (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature ψ₀) f)
      x

theorem LeanRidgelet.bochnerSynthesis_bochnerRidgelet_reluComplex_secondDifference.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {ν :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E)}
  {h : ℝ} (hh : 0 ≤ h)
  (hνp :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate
            h))
        ν =
      ν)
  (hνm :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate
            (-h)))
        ν =
      ν)
  {ψ₀ : ℝ → ℂ} (f : E → ℂ) (x : E)
  (hRplus :
    ∀
      (ξ :
        LeanRidgelet.QuadraticParameter
          E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ)
              (ψ₀
                (LeanRidgelet.quadraticArgument
                    y ξ +
                  h)))
        MeasureTheory.volume)
  (hRzero :
    ∀
      (ξ :
        LeanRidgelet.QuadraticParameter
          E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ)
              (ψ₀
                (LeanRidgelet.quadraticArgument
                  y ξ)))
        MeasureTheory.volume)
  (hRminus :
    ∀
      (ξ :
        LeanRidgelet.QuadraticParameter
          E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ)
              (ψ₀
                (LeanRidgelet.quadraticArgument
                    y ξ -
                  h)))
        MeasureTheory.volume)
  (hSplus :
    MeasureTheory.Integrable
      (fun ξ ↦
        LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              ψ₀)
            f ξ *
          LeanRidgelet.reluComplex
            (LeanRidgelet.quadraticArgument
                x ξ +
              h))
      ν)
  (hSzero :
    MeasureTheory.Integrable
      (fun ξ ↦
        LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              ψ₀)
            f ξ *
          LeanRidgelet.reluComplex
            (LeanRidgelet.quadraticArgument
              x ξ))
      ν)
  (hSminus :
    MeasureTheory.Integrable
      (fun ξ ↦
        LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              ψ₀)
            f ξ *
          LeanRidgelet.reluComplex
            (LeanRidgelet.quadraticArgument
                x ξ -
              h))
      ν) :
  LeanRidgelet.bochnerSynthesis ν
      (LeanRidgelet.quadraticVectorFeature
        LeanRidgelet.reluComplex)
      (LeanRidgelet.bochnerRidgelet
        MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature
          (LeanRidgelet.secondDifference h
            ψ₀))
        f)
      x =
    LeanRidgelet.bochnerSynthesis ν
      (LeanRidgelet.quadraticVectorFeature
        (LeanRidgelet.hatComplex h))
      (LeanRidgelet.bochnerRidgelet
        MeasureTheory.volume
        (LeanRidgelet.quadraticVectorFeature
          ψ₀)
        f)
      x

**The reduction.**  Build the second difference into the analysis feature and the composite
against the rectified linear unit becomes the composite against the hat function -- bounded,
compactly supported, of positive integral.

Two moves and one computation: the difference passes from the feature to the transform on the
analysis side, then from the coefficient function to the activation on the synthesis side, and the
second difference of the rectified linear unit *is* the hat function.  Nothing distributional
appears, because a second difference is a finite combination of translations.

This is what the remaining admissibility placeholder can be attacked through: the pair
`(reluComplex, secondDifference h ψ₀)` has the same composite as the pair `(hatComplex h, ψ₀)`, so a
probe for the second is a probe for the first, and the activation on the right is one that
Appendix C and the bounded-continuous theory can see.

theoremdefined in LeanRidgelet/HA/QuadraticSecondDifference.lean

complete

theorem LeanRidgelet.not_bochnerSynthesis_bochnerRidgelet_reluComplex_ae_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  {ν : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E)}
  {h : ℝ} (hh : 0 ≤ h)
  (hνp :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate h)) ν =
      ν)
  (hνm :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate (-h))) ν =
      ν)
  {ψ₀ : ℝ → ℂ} (f : E → ℂ)
  (hRplus :
    ∀ (ξ : LeanRidgelet.QuadraticParameter E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ)
              (ψ₀ (LeanRidgelet.quadraticArgument y ξ + h)))
        MeasureTheory.volume)
  (hRzero :
    ∀ (ξ : LeanRidgelet.QuadraticParameter E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ) (ψ₀ (LeanRidgelet.quadraticArgument y ξ)))
        MeasureTheory.volume)
  (hRminus :
    ∀ (ξ : LeanRidgelet.QuadraticParameter E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ)
              (ψ₀ (LeanRidgelet.quadraticArgument y ξ - h)))
        MeasureTheory.volume)
  (hSplus :
    ∀ (x : E),
      MeasureTheory.Integrable
        (fun ξ ↦
          LeanRidgelet.bochnerRidgelet MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature ψ₀) f ξ *
            LeanRidgelet.reluComplex
              (LeanRidgelet.quadraticArgument x ξ + h))
        ν)
  (hSzero :
    ∀ (x : E),
      MeasureTheory.Integrable
        (fun ξ ↦
          LeanRidgelet.bochnerRidgelet MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature ψ₀) f ξ *
            LeanRidgelet.reluComplex
              (LeanRidgelet.quadraticArgument x ξ))
        ν)
  (hSminus :
    ∀ (x : E),
      MeasureTheory.Integrable
        (fun ξ ↦
          LeanRidgelet.bochnerRidgelet MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature ψ₀) f ξ *
            LeanRidgelet.reluComplex
              (LeanRidgelet.quadraticArgument x ξ - h))
        ν)
  (hhat :
    ¬LeanRidgelet.bochnerSynthesis ν
          (LeanRidgelet.quadraticVectorFeature
            (LeanRidgelet.hatComplex h))
          (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature ψ₀)
            f) =ᵐ[MeasureTheory.volume]
        0) :
  ¬LeanRidgelet.bochnerSynthesis ν
        (LeanRidgelet.quadraticVectorFeature LeanRidgelet.reluComplex)
        (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
          (LeanRidgelet.quadraticVectorFeature
            (LeanRidgelet.secondDifference h ψ₀))
          f) =ᵐ[MeasureTheory.volume]
      0

theorem LeanRidgelet.not_bochnerSynthesis_bochnerRidgelet_reluComplex_ae_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  {ν :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E)}
  {h : ℝ} (hh : 0 ≤ h)
  (hνp :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate
            h))
        ν =
      ν)
  (hνm :
    MeasureTheory.Measure.map
        (⇑(LeanRidgelet.quadraticConstTranslate
            (-h)))
        ν =
      ν)
  {ψ₀ : ℝ → ℂ} (f : E → ℂ)
  (hRplus :
    ∀
      (ξ :
        LeanRidgelet.QuadraticParameter
          E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ)
              (ψ₀
                (LeanRidgelet.quadraticArgument
                    y ξ +
                  h)))
        MeasureTheory.volume)
  (hRzero :
    ∀
      (ξ :
        LeanRidgelet.QuadraticParameter
          E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ)
              (ψ₀
                (LeanRidgelet.quadraticArgument
                  y ξ)))
        MeasureTheory.volume)
  (hRminus :
    ∀
      (ξ :
        LeanRidgelet.QuadraticParameter
          E),
      MeasureTheory.Integrable
        (fun y ↦
          f y *
            (starRingEnd ℂ)
              (ψ₀
                (LeanRidgelet.quadraticArgument
                    y ξ -
                  h)))
        MeasureTheory.volume)
  (hSplus :
    ∀ (x : E),
      MeasureTheory.Integrable
        (fun ξ ↦
          LeanRidgelet.bochnerRidgelet
              MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature
                ψ₀)
              f ξ *
            LeanRidgelet.reluComplex
              (LeanRidgelet.quadraticArgument
                  x ξ +
                h))
        ν)
  (hSzero :
    ∀ (x : E),
      MeasureTheory.Integrable
        (fun ξ ↦
          LeanRidgelet.bochnerRidgelet
              MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature
                ψ₀)
              f ξ *
            LeanRidgelet.reluComplex
              (LeanRidgelet.quadraticArgument
                x ξ))
        ν)
  (hSminus :
    ∀ (x : E),
      MeasureTheory.Integrable
        (fun ξ ↦
          LeanRidgelet.bochnerRidgelet
              MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature
                ψ₀)
              f ξ *
            LeanRidgelet.reluComplex
              (LeanRidgelet.quadraticArgument
                  x ξ -
                h))
        ν)
  (hhat :
    ¬LeanRidgelet.bochnerSynthesis ν
          (LeanRidgelet.quadraticVectorFeature
            (LeanRidgelet.hatComplex h))
          (LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              ψ₀)
            f) =ᵐ[MeasureTheory.volume]
        0) :
  ¬LeanRidgelet.bochnerSynthesis ν
        (LeanRidgelet.quadraticVectorFeature
          LeanRidgelet.reluComplex)
        (LeanRidgelet.bochnerRidgelet
          MeasureTheory.volume
          (LeanRidgelet.quadraticVectorFeature
            (LeanRidgelet.secondDifference
              h ψ₀))
          f) =ᵐ[MeasureTheory.volume]
      0

**The probe transfers.**  If the composite of the hat function with an analysis feature does not
vanish on a datum, neither does the composite of the rectified linear unit with the differenced
analysis feature.  This is the nonvanishing half of admissibility, moved from an activation of
polynomial growth to a bounded compactly supported one.

defdefined in LeanRidgelet/HA/QuadraticFiniteWidth.lean

complete

def LeanRidgelet.quadraticHatFeatureBoundedContinuous.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (h : ℝ) (ξ : LeanRidgelet.QuadraticParameter E) :
  BoundedContinuousFunction E ℂ

def LeanRidgelet.quadraticHatFeatureBoundedContinuous.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ)
  (ξ :
    LeanRidgelet.QuadraticParameter E) :
  BoundedContinuousFunction E ℂ

Implementation after :=

:=
  quadraticFeatureBoundedContinuous (continuous_hatComplex h) (norm_hatComplex_le h) ξ

**The reduced activation as a bounded continuous feature.**  The hat function of
`LeanRidgelet.HA.SecondDifference` -- the second difference of the rectified linear unit -- is
continuous and bounded, so the feature it defines is an element of the space whose norm is the
supremum norm.  An activation of polynomial growth is not; differencing it twice is what makes the
finite-width theorem applicable, and it costs only a change of analysis feature.

theoremdefined in LeanRidgelet/HA/QuadraticFiniteWidth.lean

complete

theorem LeanRidgelet.quadraticHatFeatureBoundedContinuous_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ)
  (ξ : LeanRidgelet.QuadraticParameter E) (x : E) :
  (LeanRidgelet.quadraticHatFeatureBoundedContinuous h ξ) x =
    LeanRidgelet.quadraticVectorFeature (LeanRidgelet.hatComplex h) x ξ

theorem LeanRidgelet.quadraticHatFeatureBoundedContinuous_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ)
  (ξ : LeanRidgelet.QuadraticParameter E)
  (x : E) :
  (LeanRidgelet.quadraticHatFeatureBoundedContinuous
        h ξ)
      x =
    LeanRidgelet.quadraticVectorFeature
      (LeanRidgelet.hatComplex h) x ξ

theoremdefined in LeanRidgelet/HA/QuadraticFiniteWidth.lean

complete

theorem LeanRidgelet.exists_finite_quadraticNetwork_approx_hat.{u_1, u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ) {Ξ : Type u_2} [MetricSpace Ξ]
  [CompactSpace Ξ] [MeasurableSpace Ξ] [BorelSpace Ξ]
  (κ : MeasureTheory.Measure Ξ) [MeasureTheory.IsFiniteMeasure κ]
  (ι : Ξ → LeanRidgelet.QuadraticParameter E) (γ : Ξ → ℂ) {L : NNReal}
  (hlip :
    LipschitzWith L fun t ↦
      γ t • LeanRidgelet.quadraticHatFeatureBoundedContinuous h (ι t))
  {ε : ℝ} (hε : 0 < ε) :
  ∃ n w t,
    (∀ (i : Fin n), 0 ≤ w i) ∧
      ∀ (x : E),
        ‖∫ (s : Ξ),
                γ s •
                  LeanRidgelet.quadraticVectorFeature
                    (LeanRidgelet.hatComplex h) x (ι s) ∂κ -
              ∑ i,
                w i •
                  γ (t i) •
                    LeanRidgelet.quadraticVectorFeature
                      (LeanRidgelet.hatComplex h) x (ι (t i))‖ <
          ε

theorem LeanRidgelet.exists_finite_quadraticNetwork_approx_hat.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (h : ℝ)
  {Ξ : Type u_2} [MetricSpace Ξ]
  [CompactSpace Ξ] [MeasurableSpace Ξ]
  [BorelSpace Ξ]
  (κ : MeasureTheory.Measure Ξ)
  [MeasureTheory.IsFiniteMeasure κ]
  (ι :
    Ξ → LeanRidgelet.QuadraticParameter E)
  (γ : Ξ → ℂ) {L : NNReal}
  (hlip :
    LipschitzWith L fun t ↦
      γ t •
        LeanRidgelet.quadraticHatFeatureBoundedContinuous
          h (ι t))
  {ε : ℝ} (hε : 0 < ε) :
  ∃ n w t,
    (∀ (i : Fin n), 0 ≤ w i) ∧
      ∀ (x : E),
        ‖∫ (s : Ξ),
                γ s •
                  LeanRidgelet.quadraticVectorFeature
                    (LeanRidgelet.hatComplex
                      h)
                    x (ι s) ∂κ -
              ∑ i,
                w i •
                  γ (t i) •
                    LeanRidgelet.quadraticVectorFeature
                      (LeanRidgelet.hatComplex
                        h)
                      x (ι (t i))‖ <
          ε

**Appendix C for the reduced activation.**  The finite-width approximation applies to the hat
function, hence -- through the reduction of
`LeanRidgelet.bochnerSynthesis_bochnerRidgelet_reluComplex_secondDifference` -- to a network whose
activation is the rectified linear unit, once the second difference is built into the analysis
feature.

Two ways to kill the second-order pole of the rectified linear unit's transform at the origin, and only one of them works. In the convention \hat g(\zeta)=\int g(z)e^{-iz\zeta}dz, the second derivative multiplies the transform by -\zeta^2 and the second difference at step h multiplies it by -4\sin^2(h\zeta/2). Both vanish to second order at the origin. But \zeta^2 grows at infinity, and \partial_z^2 of the unit is the Dirac measure rather than a function; 4\sin^2(h\zeta/2) stays bounded by 4. The difference is the bounded multiplier with a double zero, which is exactly what a double pole needs.

Applied to the unit the difference gives an explicit function: \Delta_h^2\sigma(z)=\max(0,\,h-|z|), the hat of width 2h and height h. It is continuous, supported in [-h,h], and of positive integral — the value being h^2, the double zero cancelling the double pole and leaving a nonzero constant behind, which is the admissibility constant in its most concrete form.

The difference then moves where it has to. On the analysis side it passes from the feature to the transform, since the constant coefficient enters the feature additively and the difference acts on the data integral's integrand pointwise. On the synthesis side it passes from the coefficient function to the activation, by summation by parts: three translations and three changes of variable, needing only that the parameter measure is invariant under a translation of the constant coefficient — which the factorization above supplies. And because the action is a shear in that coefficient, the difference commutes with the group action, so the reduction respects equivariance; that statement is the difference analogue of the derivative one, and easier, no differentiability being involved.

So with \psi:=\Delta_h^2\psi_0 the composite against the rectified linear unit equals the composite against the hat function with the unmodified analysis feature. A probe for the second is a probe for the first, which moves the nonvanishing half of admissibility from an activation of polynomial growth to a bounded compactly supported one. The finite-width approximation of Appendix C also becomes available, its two conditions — continuity and boundedness — now being met.

What the reduction does not do is remove the compactness obstruction: if the composite kernel of the reduced pair is square integrable its constant still vanishes. But both features are now ordinary functions, so the failure of square integrability has to come from the parameter measure being infinite, which is the classical situation of the L1 theory's admissible pairs rather than anything special to this activation. One caveat is worth stating: 4\sin^2(h\zeta/2) also vanishes on \frac{2\pi}{h}\mathbb{Z}, so a single second difference annihilates those frequencies — harmless for a single constant, but an inversion valid for every datum would want an average over h or a difference whose multiplier vanishes only at the origin.

Theorem4.5.16

Statement uses 2

Statement dependency previews

Preview

Theorem 4.5.5

Loading preview

Statement dependency preview content is loaded from the rendered-fragment cache.

used by 0⚠L∃∀N

A nonzero reconstruction constant. The Schur scalar is named and computed by any probe, so nonvanishing reduces to one datum having nonzero image; the two bounds through the intermediate space compose to a bound on the composite; and with an admissible pair the composite returns every datum multiplied by a nonzero constant.

Lean code for Theorem4.5.16●23 declarations, 1 incomplete

abbrevdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

abbrev LeanRidgelet.QuadraticMachine.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] : Type u_1

abbrev LeanRidgelet.QuadraticMachine.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] : Type u_1

Implementation after :=

:=
  JointEquivariantMachine
    (quadraticRelativeParameterLpUnitaryRepresentation lam).toContRepresentation
    (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E)).toContRepresentation

A bounded machine for the quadratic-form network: an intertwiner from the relatively invariant
parameter representation to the affine data representation.

abbrevdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

abbrev LeanRidgelet.QuadraticRidgelet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] : Type u_1

abbrev LeanRidgelet.QuadraticRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] : Type u_1

Implementation after :=

:=
  JointEquivariantRidgelet
    (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E)).toContRepresentation
    (quadraticRelativeParameterLpUnitaryRepresentation lam).toContRepresentation

A bounded ridgelet transform for the quadratic-form network: an intertwiner the other way.

defdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

def LeanRidgelet.quadraticReconstructionScalar.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (M : LeanRidgelet.QuadraticMachine lam)
  (R : LeanRidgelet.QuadraticRidgelet lam) : ℂ

def LeanRidgelet.quadraticReconstructionScalar.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M : LeanRidgelet.QuadraticMachine lam)
  (R :
    LeanRidgelet.QuadraticRidgelet lam) :
  ℂ

Implementation after :=

:=
  (quadraticReconstructionOperator_eq_smul_id lam M R).choose

**The reconstruction scalar of a pair.**  Schur's lemma makes the reconstruction operator a
scalar multiple of the identity; this is that scalar.  It is the admissibility constant of the pair:
`LeanRidgelet.quadraticReconstructionScalar_eq_inner_div` computes it from any nonzero datum, so
nothing about the choice is arbitrary.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.quadraticReconstructionOperator_eq_scalar_smul_id.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (M : LeanRidgelet.QuadraticMachine lam)
  (R : LeanRidgelet.QuadraticRidgelet lam) :
  LeanRidgelet.jointReconstructionOperator M R =
    LeanRidgelet.quadraticReconstructionScalar lam M R •
      ContinuousLinearMap.id ℂ
        ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)

theorem LeanRidgelet.quadraticReconstructionOperator_eq_scalar_smul_id.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M : LeanRidgelet.QuadraticMachine lam)
  (R :
    LeanRidgelet.QuadraticRidgelet lam) :
  LeanRidgelet.jointReconstructionOperator
      M R =
    LeanRidgelet.quadraticReconstructionScalar
        lam M R •
      ContinuousLinearMap.id ℂ
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)

The reconstruction operator is its scalar times the identity.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.quadraticReconstruction_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (M : LeanRidgelet.QuadraticMachine lam)
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) :
  M (R f) = LeanRidgelet.quadraticReconstructionScalar lam M R • f

theorem LeanRidgelet.quadraticReconstruction_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M : LeanRidgelet.QuadraticMachine lam)
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)) :
  M (R f) =
    LeanRidgelet.quadraticReconstructionScalar
        lam M R •
      f

The pointwise form: the machine applied to the transform of a datum is the scalar times the
datum.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.quadraticReconstructionScalar_ne_zero.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (M : LeanRidgelet.QuadraticMachine lam)
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume))
  (hf : M (R f) ≠ 0) :
  LeanRidgelet.quadraticReconstructionScalar lam M R ≠ 0

theorem LeanRidgelet.quadraticReconstructionScalar_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M : LeanRidgelet.QuadraticMachine lam)
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume))
  (hf : M (R f) ≠ 0) :
  LeanRidgelet.quadraticReconstructionScalar
      lam M R ≠
    0

**One probe decides the nonvanishing.**  If some datum has nonzero image under the composite
then
the reconstruction scalar is nonzero.  This is why the whole question reduces to exhibiting a single
datum, and why the admissibility statement below is an existence.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.inner_quadraticReconstruction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (M : LeanRidgelet.QuadraticMachine lam)
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) :
  inner ℂ f (M (R f)) =
    LeanRidgelet.quadraticReconstructionScalar lam M R * inner ℂ f f

theorem LeanRidgelet.inner_quadraticReconstruction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M : LeanRidgelet.QuadraticMachine lam)
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)) :
  inner ℂ f (M (R f)) =
    LeanRidgelet.quadraticReconstructionScalar
        lam M R *
      inner ℂ f f

**The scalar is the admissibility constant.**  Every datum computes it: the pairing of the datum
with its reconstruction is the scalar times the datum's squared norm.  So the scalar is determined
by
the pair of features and by no choice made in its definition, and for a nonzero datum it is the
Rayleigh quotient of the reconstruction operator.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.quadratic_reconstruction_nonzero_of_apply_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (M : LeanRidgelet.QuadraticMachine lam)
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume))
  (hf : M (R f) ≠ 0) :
  ∃ c,
    c ≠ 0 ∧
      LeanRidgelet.jointReconstructionOperator M R =
          c •
            ContinuousLinearMap.id ℂ
              ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) ∧
        Function.RightInverse ⇑(c⁻¹ • R.toContinuousLinearMap) ⇑M

theorem LeanRidgelet.quadratic_reconstruction_nonzero_of_apply_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (M : LeanRidgelet.QuadraticMachine lam)
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume))
  (hf : M (R f) ≠ 0) :
  ∃ c,
    c ≠ 0 ∧
      LeanRidgelet.jointReconstructionOperator
            M R =
          c •
            ContinuousLinearMap.id ℂ
              ↥(MeasureTheory.Lp ℂ 2
                  MeasureTheory.volume) ∧
        Function.RightInverse
          ⇑(c⁻¹ • R.toContinuousLinearMap)
          ⇑M

**Section 7 with a nonzero reconstruction constant, from a probe.**  A pair of bounded
intertwiners and one datum whose reconstruction is nonzero give a reconstruction formula whose
constant is nonzero, so the normalized ridgelet transform is a genuine right inverse of the machine.
Nothing is assumed beyond the probe, and the proof carries no placeholder.

defdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

def LeanRidgelet.QuadraticAnalysisBound.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (ψ : ℝ → ℂ) (k : ℕ) : Prop

def LeanRidgelet.QuadraticAnalysisBound.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (ψ : ℝ → ℂ) (k : ℕ) : Prop

Implementation after :=

:=
  ∃ C : ℝ≥0∞, C ≠ ∞ ∧ ∀ f : Lp ℂ 2 (volume : Measure E),
    quadraticSobolevSeminorm lam k
        (bochnerRidgelet (volume : Measure E) (quadraticVectorFeature ψ) (f : E → ℂ)) ≤
      C * eLpNorm (f : E → ℂ) 2 (volume : Measure E)

**The analysis bound through `Γ^k`.**  The order-`k` Sobolev seminorm in the constant
coefficient
of the analysis transform of a datum is at most a finite constant times the datum's `L²` norm.  This
is the bound `LeanRidgelet.quadraticSobolevSeminorm_bochnerRidgelet` reduces to an estimate on the
derivative features, and it is where smoothness of the analysis feature is spent.

defdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

def LeanRidgelet.QuadraticSynthesisBound.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  (σ : ℝ → ℂ) (k : ℕ) : Prop

def LeanRidgelet.QuadraticSynthesisBound.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  (σ : ℝ → ℂ) (k : ℕ) : Prop

Implementation after :=

:=
  ∃ C : ℝ≥0∞, C ≠ ∞ ∧ ∀ γ : QuadraticParameter E → ℂ, MemQuadraticSobolev lam k γ →
    eLpNorm (bochnerSynthesis (quadraticRelativeMeasure lam) (quadraticVectorFeature σ) γ) 2
        (volume : Measure E) ≤
      C * quadraticSobolevSeminorm lam k γ

**The synthesis bound out of `Γ^k`.**  The `L²` norm of the synthesis integral of a coefficient
function in the space is at most a finite constant times its order-`k` seminorm.

A warning about how *not* to prove it.  The theorem
`LeanRidgelet.enorm_bochnerSynthesis_le_quadraticSobolevSeminorm_mul` gives this pointwise in the
data variable, by Cauchy--Schwarz in the parameter -- but integrating that estimate in the data
variable asks exactly for the square integrability of the feature over the product, which is
condition T2, which
`LeanRidgelet.HA.QuadraticComposite` shows forces the reconstruction constant to vanish.  The
pointwise estimate uses only the order-`0` term of the seminorm, and that is why: a bound that does
not see the derivatives cannot escape Hilbert--Schmidt.  A proof has to use the higher-order terms,
pairing the derivatives of the coefficient function against a negative-order object built from the
synthesis feature.  For an activation of polynomial growth that also needs a polynomial weight in
the constant coefficient -- the growth index `t` of the L2 activation spaces -- which `Γ^k` as
defined does not carry.

abbrevdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

abbrev LeanRidgelet.QuadraticCompositeEndomorphism.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] : Type u_1

abbrev LeanRidgelet.QuadraticCompositeEndomorphism.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] :
  Type u_1

Implementation after :=

:=
  JointEquivariantMachine
    (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E)).toContRepresentation
    (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E)).toContRepresentation

A bounded intertwining **endomorphism** of data `L²`: the shape the article's weaker boundedness
hypothesis takes, and -- as `LeanRidgelet.eLpNorm_bochnerSynthesis_bochnerRidgelet_le` shows -- the
shape the intermediate-space route actually produces.  The two bounds control the composite, not the
two factors separately: the synthesis is bounded on `Γ^k`, which is smaller than parameter `L²`, so
nothing in them says the machine is bounded there.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.eLpNorm_bochnerSynthesis_bochnerRidgelet_le.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  {σ ψ : ℝ → ℂ} {k : ℕ}
  (hψ : LeanRidgelet.QuadraticAnalysisBound lam ψ k)
  (hσ : LeanRidgelet.QuadraticSynthesisBound lam σ k) :
  ∃ C,
    C ≠ ⊤ ∧
      ∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
        LeanRidgelet.MemQuadraticSobolev lam k
            (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f) →
          MeasureTheory.eLpNorm
              (LeanRidgelet.bochnerSynthesis
                (LeanRidgelet.quadraticRelativeMeasure lam)
                (LeanRidgelet.quadraticVectorFeature σ)
                (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
                  (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f))
              2 MeasureTheory.volume ≤
            C * MeasureTheory.eLpNorm (↑↑f) 2 MeasureTheory.volume

theorem LeanRidgelet.eLpNorm_bochnerSynthesis_bochnerRidgelet_le.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  {σ ψ : ℝ → ℂ} {k : ℕ}
  (hψ :
    LeanRidgelet.QuadraticAnalysisBound
      lam ψ k)
  (hσ :
    LeanRidgelet.QuadraticSynthesisBound
      lam σ k) :
  ∃ C,
    C ≠ ⊤ ∧
      ∀
        (f :
          ↥(MeasureTheory.Lp ℂ 2
              MeasureTheory.volume)),
        LeanRidgelet.MemQuadraticSobolev
            lam k
            (LeanRidgelet.bochnerRidgelet
              MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature
                ψ)
              ↑↑f) →
          MeasureTheory.eLpNorm
              (LeanRidgelet.bochnerSynthesis
                (LeanRidgelet.quadraticRelativeMeasure
                  lam)
                (LeanRidgelet.quadraticVectorFeature
                  σ)
                (LeanRidgelet.bochnerRidgelet
                  MeasureTheory.volume
                  (LeanRidgelet.quadraticVectorFeature
                    ψ)
                  ↑↑f))
              2 MeasureTheory.volume ≤
            C *
              MeasureTheory.eLpNorm (↑↑f)
                2 MeasureTheory.volume

**The composite is bounded, and neither factor need be.**  The two bounds compose: the analysis
transform of a datum has order-`k` seminorm at most a constant times the datum's `L²` norm, and the
synthesis integral of a member of the space has `L²` norm at most a constant times its seminorm, so
the composite of the two Bochner integrals is bounded on data `L²` with the product of the two
constants.

This is the estimate the whole route exists to produce, and the point is what it does *not* say.  It
does not say the synthesis integral is bounded on parameter `L²`: it is bounded on `Γ^k`, which is
smaller, and the seminorm dominates the `L²` norm rather than the other way round.  So the composite
is bounded without either factor being Hilbert--Schmidt, which is exactly the shape
`LeanRidgelet.HA.QuadraticComposite` shows is needed -- and the shape the article's boundedness
hypothesis takes when read as a hypothesis on the composite alone.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.exists_ne_zero_of_quadraticCompositeEndomorphism.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] (T : LeanRidgelet.QuadraticCompositeEndomorphism)
  (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) (hf : T f ≠ 0) :
  ∃ c,
    c ≠ 0 ∧
      T.toContinuousLinearMap =
        c •
          ContinuousLinearMap.id ℂ
            ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)

theorem LeanRidgelet.exists_ne_zero_of_quadraticCompositeEndomorphism.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  (T :
    LeanRidgelet.QuadraticCompositeEndomorphism)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume))
  (hf : T f ≠ 0) :
  ∃ c,
    c ≠ 0 ∧
      T.toContinuousLinearMap =
        c •
          ContinuousLinearMap.id ℂ
            ↥(MeasureTheory.Lp ℂ 2
                MeasureTheory.volume)

**The endomorphism form of the endpoint, with no placeholder.**  A bounded intertwining
endomorphism of data `L²` that does not annihilate some datum is a **nonzero** scalar multiple of
the identity.  Schur gives the scalar and the probe gives its nonvanishing; this is the same
two-line reduction as for a pair of intertwiners, in the shape the intermediate-space route
produces.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.exists_quadraticCompositeIntertwiner_of_bounds.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ} {k : ℕ}
  (hψ : LeanRidgelet.QuadraticAnalysisBound lam ψ k)
  (hσ : LeanRidgelet.QuadraticSynthesisBound lam σ k)
  (hmemΓ :
    ∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      LeanRidgelet.MemQuadraticSobolev lam k
        (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
          (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f))
  (hmeas :
    ∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      MeasureTheory.AEStronglyMeasurable
        (LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure lam)
          (LeanRidgelet.quadraticVectorFeature σ)
          (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f))
        MeasureTheory.volume)
  (hadd :
    ∀ (f h : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure lam)
          (LeanRidgelet.quadraticVectorFeature σ)
          (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature ψ)
            ↑↑(f + h)) =ᵐ[MeasureTheory.volume]
        LeanRidgelet.bochnerSynthesis
            (LeanRidgelet.quadraticRelativeMeasure lam)
            (LeanRidgelet.quadraticVectorFeature σ)
            (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f) +
          LeanRidgelet.bochnerSynthesis
            (LeanRidgelet.quadraticRelativeMeasure lam)
            (LeanRidgelet.quadraticVectorFeature σ)
            (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature ψ) ↑↑h))
  (hsmul :
    ∀ (c : ℂ) (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure lam)
          (LeanRidgelet.quadraticVectorFeature σ)
          (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature ψ)
            ↑↑(c • f)) =ᵐ[MeasureTheory.volume]
        c •
          LeanRidgelet.bochnerSynthesis
            (LeanRidgelet.quadraticRelativeMeasure lam)
            (LeanRidgelet.quadraticVectorFeature σ)
            (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f)) :
  ∃ T,
    ∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      ↑↑(T f) =ᵐ[MeasureTheory.volume]
        LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure lam)
          (LeanRidgelet.quadraticVectorFeature σ)
          (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f)

theorem LeanRidgelet.exists_quadraticCompositeIntertwiner_of_bounds.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  {k : ℕ}
  (hψ :
    LeanRidgelet.QuadraticAnalysisBound
      lam ψ k)
  (hσ :
    LeanRidgelet.QuadraticSynthesisBound
      lam σ k)
  (hmemΓ :
    ∀
      (f :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      LeanRidgelet.MemQuadraticSobolev lam
        k
        (LeanRidgelet.bochnerRidgelet
          MeasureTheory.volume
          (LeanRidgelet.quadraticVectorFeature
            ψ)
          ↑↑f))
  (hmeas :
    ∀
      (f :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      MeasureTheory.AEStronglyMeasurable
        (LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure
            lam)
          (LeanRidgelet.quadraticVectorFeature
            σ)
          (LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              ψ)
            ↑↑f))
        MeasureTheory.volume)
  (hadd :
    ∀
      (f h :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure
            lam)
          (LeanRidgelet.quadraticVectorFeature
            σ)
          (LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              ψ)
            ↑↑(f +
                  h)) =ᵐ[MeasureTheory.volume]
        LeanRidgelet.bochnerSynthesis
            (LeanRidgelet.quadraticRelativeMeasure
              lam)
            (LeanRidgelet.quadraticVectorFeature
              σ)
            (LeanRidgelet.bochnerRidgelet
              MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature
                ψ)
              ↑↑f) +
          LeanRidgelet.bochnerSynthesis
            (LeanRidgelet.quadraticRelativeMeasure
              lam)
            (LeanRidgelet.quadraticVectorFeature
              σ)
            (LeanRidgelet.bochnerRidgelet
              MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature
                ψ)
              ↑↑h))
  (hsmul :
    ∀ (c : ℂ)
      (f :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure
            lam)
          (LeanRidgelet.quadraticVectorFeature
            σ)
          (LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              ψ)
            ↑↑(c •
                  f)) =ᵐ[MeasureTheory.volume]
        c •
          LeanRidgelet.bochnerSynthesis
            (LeanRidgelet.quadraticRelativeMeasure
              lam)
            (LeanRidgelet.quadraticVectorFeature
              σ)
            (LeanRidgelet.bochnerRidgelet
              MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature
                ψ)
              ↑↑f)) :
  ∃ T,
    ∀
      (f :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      ↑↑(T f) =ᵐ[MeasureTheory.volume]
        LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure
            lam)
          (LeanRidgelet.quadraticVectorFeature
            σ)
          (LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              ψ)
            ↑↑f)

**The composite is a bounded intertwining endomorphism.**  Given the two bounds through `Γ^k`,
the integrability that makes the two Bochner integrals additive and homogeneous on almost-everywhere
classes, and measurability of the composite, the composite is realized by a bounded intertwining
endomorphism of data `L²`.

Every hypothesis beyond the two bounds is one that packaging a pointwise formula as an operator
requires and that no estimate can supply.  A formula linear on functions need not be linear on
almost-everywhere classes: splitting the defining integral over a sum needs each piece integrable,
so `hadd` and `hsmul` are where the convergence of the two integrals enters.  Membership of the
analysis transform in the space is what the synthesis bound is stated against, and measurability is
what turns a finite norm into membership of `L²`.

The estimate is `LeanRidgelet.eLpNorm_bochnerSynthesis_bochnerRidgelet_le`, the packaging is
`MeasureTheory.lpOperatorOfPointwise`, and the equivariance is
`LeanRidgelet.quadraticComposite_intertwines_of_coeFn` -- which uses nothing about how the operator
was built.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

contains sorry

theorem LeanRidgelet.exists_quadraticAdmissiblePair.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] :
  ∃ σ ψ k,
    LeanRidgelet.QuadraticAnalysisBound lam ψ k ∧
      LeanRidgelet.QuadraticSynthesisBound lam σ k ∧
        (∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
            LeanRidgelet.MemQuadraticSobolev lam k
              (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
                (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f)) ∧
          (∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
              MeasureTheory.AEStronglyMeasurable
                (LeanRidgelet.bochnerSynthesis
                  (LeanRidgelet.quadraticRelativeMeasure lam)
                  (LeanRidgelet.quadraticVectorFeature σ)
                  (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
                    (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f))
                MeasureTheory.volume) ∧
            (∀ (f h : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
                LeanRidgelet.bochnerSynthesis
                    (LeanRidgelet.quadraticRelativeMeasure lam)
                    (LeanRidgelet.quadraticVectorFeature σ)
                    (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
                      (LeanRidgelet.quadraticVectorFeature ψ)
                      ↑↑(f + h)) =ᵐ[MeasureTheory.volume]
                  LeanRidgelet.bochnerSynthesis
                      (LeanRidgelet.quadraticRelativeMeasure lam)
                      (LeanRidgelet.quadraticVectorFeature σ)
                      (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
                        (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f) +
                    LeanRidgelet.bochnerSynthesis
                      (LeanRidgelet.quadraticRelativeMeasure lam)
                      (LeanRidgelet.quadraticVectorFeature σ)
                      (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
                        (LeanRidgelet.quadraticVectorFeature ψ) ↑↑h)) ∧
              (∀ (c : ℂ)
                  (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
                  LeanRidgelet.bochnerSynthesis
                      (LeanRidgelet.quadraticRelativeMeasure lam)
                      (LeanRidgelet.quadraticVectorFeature σ)
                      (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
                        (LeanRidgelet.quadraticVectorFeature ψ)
                        ↑↑(c • f)) =ᵐ[MeasureTheory.volume]
                    c •
                      LeanRidgelet.bochnerSynthesis
                        (LeanRidgelet.quadraticRelativeMeasure lam)
                        (LeanRidgelet.quadraticVectorFeature σ)
                        (LeanRidgelet.bochnerRidgelet
                          MeasureTheory.volume
                          (LeanRidgelet.quadraticVectorFeature ψ)
                          ↑↑f)) ∧
                ∃ f,
                  ¬LeanRidgelet.bochnerSynthesis
                        (LeanRidgelet.quadraticRelativeMeasure lam)
                        (LeanRidgelet.quadraticVectorFeature σ)
                        (LeanRidgelet.bochnerRidgelet
                          MeasureTheory.volume
                          (LeanRidgelet.quadraticVectorFeature ψ)
                          ↑↑f) =ᵐ[MeasureTheory.volume]
                      0

theorem LeanRidgelet.exists_quadraticAdmissiblePair.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] :
  ∃ σ ψ k,
    LeanRidgelet.QuadraticAnalysisBound
        lam ψ k ∧
      LeanRidgelet.QuadraticSynthesisBound
          lam σ k ∧
        (∀
            (f :
              ↥(MeasureTheory.Lp ℂ 2
                  MeasureTheory.volume)),
            LeanRidgelet.MemQuadraticSobolev
              lam k
              (LeanRidgelet.bochnerRidgelet
                MeasureTheory.volume
                (LeanRidgelet.quadraticVectorFeature
                  ψ)
                ↑↑f)) ∧
          (∀
              (f :
                ↥(MeasureTheory.Lp ℂ 2
                    MeasureTheory.volume)),
              MeasureTheory.AEStronglyMeasurable
                (LeanRidgelet.bochnerSynthesis
                  (LeanRidgelet.quadraticRelativeMeasure
                    lam)
                  (LeanRidgelet.quadraticVectorFeature
                    σ)
                  (LeanRidgelet.bochnerRidgelet
                    MeasureTheory.volume
                    (LeanRidgelet.quadraticVectorFeature
                      ψ)
                    ↑↑f))
                MeasureTheory.volume) ∧
            (∀
                (f h :
                  ↥(MeasureTheory.Lp ℂ 2
                      MeasureTheory.volume)),
                LeanRidgelet.bochnerSynthesis
                    (LeanRidgelet.quadraticRelativeMeasure
                      lam)
                    (LeanRidgelet.quadraticVectorFeature
                      σ)
                    (LeanRidgelet.bochnerRidgelet
                      MeasureTheory.volume
                      (LeanRidgelet.quadraticVectorFeature
                        ψ)
                      ↑↑(f +
                            h)) =ᵐ[MeasureTheory.volume]
                  LeanRidgelet.bochnerSynthesis
                      (LeanRidgelet.quadraticRelativeMeasure
                        lam)
                      (LeanRidgelet.quadraticVectorFeature
                        σ)
                      (LeanRidgelet.bochnerRidgelet
                        MeasureTheory.volume
                        (LeanRidgelet.quadraticVectorFeature
                          ψ)
                        ↑↑f) +
                    LeanRidgelet.bochnerSynthesis
                      (LeanRidgelet.quadraticRelativeMeasure
                        lam)
                      (LeanRidgelet.quadraticVectorFeature
                        σ)
                      (LeanRidgelet.bochnerRidgelet
                        MeasureTheory.volume
                        (LeanRidgelet.quadraticVectorFeature
                          ψ)
                        ↑↑h)) ∧
              (∀ (c : ℂ)
                  (f :
                    ↥(MeasureTheory.Lp ℂ 2
                        MeasureTheory.volume)),
                  LeanRidgelet.bochnerSynthesis
                      (LeanRidgelet.quadraticRelativeMeasure
                        lam)
                      (LeanRidgelet.quadraticVectorFeature
                        σ)
                      (LeanRidgelet.bochnerRidgelet
                        MeasureTheory.volume
                        (LeanRidgelet.quadraticVectorFeature
                          ψ)
                        ↑↑(c •
                              f)) =ᵐ[MeasureTheory.volume]
                    c •
                      LeanRidgelet.bochnerSynthesis
                        (LeanRidgelet.quadraticRelativeMeasure
                          lam)
                        (LeanRidgelet.quadraticVectorFeature
                          σ)
                        (LeanRidgelet.bochnerRidgelet
                          MeasureTheory.volume
                          (LeanRidgelet.quadraticVectorFeature
                            ψ)
                          ↑↑f)) ∧
                ∃ f,
                  ¬LeanRidgelet.bochnerSynthesis
                        (LeanRidgelet.quadraticRelativeMeasure
                          lam)
                        (LeanRidgelet.quadraticVectorFeature
                          σ)
                        (LeanRidgelet.bochnerRidgelet
                          MeasureTheory.volume
                          (LeanRidgelet.quadraticVectorFeature
                            ψ)
                          ↑↑f) =ᵐ[MeasureTheory.volume]
                      0

**Remaining: an admissible pair exists.**  Some synthesis feature, analysis feature and order
satisfy both bounds and leave the pointwise composite nonvanishing on some datum.  This is the
article's admissibility constant being nonzero.

Not proved here, and stated as an existence over pairs on purpose: for a fixed pair the conclusion
is false, the zero synthesis feature satisfying both bounds and annihilating every datum.  The
article fixes the activation to be the rectified linear unit; specializing this existence to a fixed
activation needs its nonvanishing on the Fourier side together with the negative-order condition
that controls its growth, and neither is assumed anywhere in this development.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.quadratic_reconstruction_nonzero.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] :
  ∃ σ ψ k c,
    c ≠ 0 ∧
      LeanRidgelet.QuadraticAnalysisBound lam ψ k ∧
        LeanRidgelet.QuadraticSynthesisBound lam σ k ∧
          ∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
            LeanRidgelet.bochnerSynthesis
                (LeanRidgelet.quadraticRelativeMeasure lam)
                (LeanRidgelet.quadraticVectorFeature σ)
                (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
                  (LeanRidgelet.quadraticVectorFeature ψ)
                  ↑↑f) =ᵐ[MeasureTheory.volume]
              fun x ↦ c • ↑↑f x

theorem LeanRidgelet.quadratic_reconstruction_nonzero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] :
  ∃ σ ψ k c,
    c ≠ 0 ∧
      LeanRidgelet.QuadraticAnalysisBound
          lam ψ k ∧
        LeanRidgelet.QuadraticSynthesisBound
            lam σ k ∧
          ∀
            (f :
              ↥(MeasureTheory.Lp ℂ 2
                  MeasureTheory.volume)),
            LeanRidgelet.bochnerSynthesis
                (LeanRidgelet.quadraticRelativeMeasure
                  lam)
                (LeanRidgelet.quadraticVectorFeature
                  σ)
                (LeanRidgelet.bochnerRidgelet
                  MeasureTheory.volume
                  (LeanRidgelet.quadraticVectorFeature
                    ψ)
                  ↑↑f) =ᵐ[MeasureTheory.volume]
              fun x ↦ c • ↑↑f x

**Section 7 of the article, with a nonzero reconstruction constant.**  There are a synthesis
feature, an analysis feature, an order satisfying both bounds through the intermediate coefficient
space, and a **nonzero** constant such that the composite of the two Bochner integrals returns every
datum multiplied by that constant.  Equivalently: the network whose coefficient function is the
normalized analysis transform of a datum outputs that datum.  This is the universality claim for the
quadratic-form network, stated on the network's own integrals rather than on an abstract pair of
operators.

The Schur step and the reduction of nonvanishing to a single probe carry nothing, and the composite
estimate is proved.  What this rests on is exactly the two placeholders above,
`LeanRidgelet.exists_quadraticCompositeIntertwiner_of_bounds` and
`LeanRidgelet.exists_quadraticAdmissiblePair`.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.quadratic_reconstruction_adjoint.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (R : LeanRidgelet.QuadraticRidgelet lam)
  (f₀ : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) (hf₀ : R f₀ ≠ 0) :
  ∃ M c,
    c ≠ 0 ∧
      LeanRidgelet.jointReconstructionOperator M R =
          c •
            ContinuousLinearMap.id ℂ
              ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) ∧
        Function.RightInverse ⇑(c⁻¹ • R.toContinuousLinearMap) ⇑M ∧
          ∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
            inner ℂ (R f) (R f) = c * inner ℂ f f

theorem LeanRidgelet.quadratic_reconstruction_adjoint.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure]
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (f₀ :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume))
  (hf₀ : R f₀ ≠ 0) :
  ∃ M c,
    c ≠ 0 ∧
      LeanRidgelet.jointReconstructionOperator
            M R =
          c •
            ContinuousLinearMap.id ℂ
              ↥(MeasureTheory.Lp ℂ 2
                  MeasureTheory.volume) ∧
        Function.RightInverse
            ⇑(c⁻¹ •
                R.toContinuousLinearMap)
            ⇑M ∧
          ∀
            (f :
              ↥(MeasureTheory.Lp ℂ 2
                  MeasureTheory.volume)),
            inner ℂ (R f) (R f) =
              c * inner ℂ f f

**Section 7 with a nonzero reconstruction constant, unconditionally.**  Take the machine to be
the adjoint of the ridgelet transform.  Then a single datum with nonzero transform gives a
**nonzero** reconstruction constant, the reconstruction operator is that constant times the
identity, and the normalized ridgelet transform is a right inverse of the machine -- the article's
endpoint, with nothing left assumed but the boundedness of the ridgelet transform itself.

This is the argument of coorbit theory and of the theory of generalized wavelet transforms; see
`LeanRidgelet.HA.AdjointReconstruction` for the general statement and for the reference.  The
constant is `‖R f‖² / ‖f‖²`, which the orthogonality relation below exhibits, so it is real and
positive rather than merely nonzero, and the compactness obstruction of
`LeanRidgelet.HA.QuadraticComposite` cannot arise: no kernel is assumed square integrable, and the
composite is a multiple of an isometry, which in infinite dimensions is not compact.

What it costs is that the machine is the adjoint rather than an independently chosen synthesis
integral.  At the level of the Bochner formulas that is the common-feature case -- the adjoint of
the ridgelet transform against a feature is the synthesis integral against the same feature -- so
this reconstructs with the network whose activation is the analysis feature.  Identifying the
adjoint with the pointwise synthesis integral is a Fubini computation and is not done here; fixing
the activation in advance is the harder problem the two placeholders above are about.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.quadratic_bochner_reconstruction_adjoint.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (M : LeanRidgelet.QuadraticMachine lam)
  (hR :
    ∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      ↑↑(R f) =ᵐ[LeanRidgelet.quadraticRelativeMeasure lam]
        LeanRidgelet.bochnerRidgelet MeasureTheory.volume
          (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f)
  (hM :
    ∀
      (γ :
        ↥(MeasureTheory.Lp ℂ 2
            (LeanRidgelet.quadraticRelativeMeasure lam))),
      ↑↑(M γ) =ᵐ[MeasureTheory.volume]
        LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure lam)
          (LeanRidgelet.quadraticVectorFeature ψ) ↑↑γ)
  (hfub :
    ∀
      (γ :
        ↥(MeasureTheory.Lp ℂ 2
            (LeanRidgelet.quadraticRelativeMeasure lam)))
      (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      MeasureTheory.Integrable
        (Function.uncurry fun x ξ ↦
          (starRingEnd ℂ) (↑↑γ ξ) *
            ((starRingEnd ℂ)
                (LeanRidgelet.quadraticVectorFeature ψ x ξ) *
              ↑↑f x))
        (MeasureTheory.volume.prod
          (LeanRidgelet.quadraticRelativeMeasure lam)))
  (f₀ : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) (hf₀ : R f₀ ≠ 0) :
  ∃ c,
    c ≠ 0 ∧
      ∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
        LeanRidgelet.bochnerSynthesis
            (LeanRidgelet.quadraticRelativeMeasure lam)
            (LeanRidgelet.quadraticVectorFeature ψ)
            (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature ψ)
              ↑↑f) =ᵐ[MeasureTheory.volume]
          fun x ↦ c • ↑↑f x

theorem LeanRidgelet.quadratic_bochner_reconstruction_adjoint.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {ψ : ℝ → ℂ}
  (R : LeanRidgelet.QuadraticRidgelet lam)
  (M : LeanRidgelet.QuadraticMachine lam)
  (hR :
    ∀
      (f :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      ↑↑(R
              f) =ᵐ[LeanRidgelet.quadraticRelativeMeasure
          lam]
        LeanRidgelet.bochnerRidgelet
          MeasureTheory.volume
          (LeanRidgelet.quadraticVectorFeature
            ψ)
          ↑↑f)
  (hM :
    ∀
      (γ :
        ↥(MeasureTheory.Lp ℂ 2
            (LeanRidgelet.quadraticRelativeMeasure
              lam))),
      ↑↑(M γ) =ᵐ[MeasureTheory.volume]
        LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure
            lam)
          (LeanRidgelet.quadraticVectorFeature
            ψ)
          ↑↑γ)
  (hfub :
    ∀
      (γ :
        ↥(MeasureTheory.Lp ℂ 2
            (LeanRidgelet.quadraticRelativeMeasure
              lam)))
      (f :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      MeasureTheory.Integrable
        (Function.uncurry fun x ξ ↦
          (starRingEnd ℂ) (↑↑γ ξ) *
            ((starRingEnd ℂ)
                (LeanRidgelet.quadraticVectorFeature
                  ψ x ξ) *
              ↑↑f x))
        (MeasureTheory.volume.prod
          (LeanRidgelet.quadraticRelativeMeasure
            lam)))
  (f₀ :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume))
  (hf₀ : R f₀ ≠ 0) :
  ∃ c,
    c ≠ 0 ∧
      ∀
        (f :
          ↥(MeasureTheory.Lp ℂ 2
              MeasureTheory.volume)),
        LeanRidgelet.bochnerSynthesis
            (LeanRidgelet.quadraticRelativeMeasure
              lam)
            (LeanRidgelet.quadraticVectorFeature
              ψ)
            (LeanRidgelet.bochnerRidgelet
              MeasureTheory.volume
              (LeanRidgelet.quadraticVectorFeature
                ψ)
              ↑↑f) =ᵐ[MeasureTheory.volume]
          fun x ↦ c • ↑↑f x

**The quadratic-form network reconstructs, on its own integrals.**  Take the synthesis feature
to be the analysis feature.  Then the synthesis integral is the adjoint of the ridgelet transform,
so one datum with nonzero transform gives a **nonzero** constant for which the composite of the
network's two Bochner integrals returns every datum multiplied by that constant.

This is the Section 7 endpoint stated on the integrals the article writes down rather than on
abstract operators, and it carries no placeholder.  What is assumed is what the development assumes
everywhere for analytic input -- that the two integrals are realized by bounded intertwiners --
together with one Fubini hypothesis and the nonvanishing probe.  The constant is `‖R f‖² / ‖f‖²`, so
it is positive.

The synthesis feature being the analysis feature is what makes the two integrals adjoint, and it is
the whole cost of the route: this reconstructs with the network whose activation is the analysis
feature.  An activation fixed in advance is the harder problem of the two placeholders above.

abbrevdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

abbrev LeanRidgelet.QuadraticGammaMachine.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) : Type u_1

abbrev LeanRidgelet.QuadraticGammaMachine.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) :
  Type u_1

Implementation after :=

:=
  JointEquivariantMachine (quadraticSobolevContRepresentation lam k)
    (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E)).toContRepresentation

A machine out of the intermediate coefficient space: an intertwiner from the representation on
`Γ^k` to the affine data representation.  This is the shape the two bounds through `Γ^k` control --
the synthesis is bounded on `Γ^k`, not on parameter `L²` -- so it is the shape in which a pair of
intertwiners is available at all.

abbrevdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

abbrev LeanRidgelet.QuadraticGammaRidgelet.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) : Type u_1

abbrev LeanRidgelet.QuadraticGammaRidgelet.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ) :
  Type u_1

Implementation after :=

:=
  JointEquivariantRidgelet
    (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E)).toContRepresentation
    (quadraticSobolevContRepresentation lam k)

A ridgelet transform into the intermediate coefficient space.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.quadratic_gamma_reconstruction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (M : LeanRidgelet.QuadraticGammaMachine lam k)
  (R : LeanRidgelet.QuadraticGammaRidgelet lam k) :
  ∃ c,
    LeanRidgelet.jointReconstructionOperator M R =
        c •
          ContinuousLinearMap.id ℂ
            ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) ∧
      (c ≠ 0 →
        Function.RightInverse ⇑(c⁻¹ • R.toContinuousLinearMap) ⇑M)

theorem LeanRidgelet.quadratic_gamma_reconstruction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (M :
    LeanRidgelet.QuadraticGammaMachine lam
      k)
  (R :
    LeanRidgelet.QuadraticGammaRidgelet
      lam k) :
  ∃ c,
    LeanRidgelet.jointReconstructionOperator
          M R =
        c •
          ContinuousLinearMap.id ℂ
            ↥(MeasureTheory.Lp ℂ 2
                MeasureTheory.volume) ∧
      (c ≠ 0 →
        Function.RightInverse
          ⇑(c⁻¹ • R.toContinuousLinearMap)
          ⇑M)

**Section 7 through the intermediate space.**  A machine out of `Γ^k` and a ridgelet transform
into it compose to a scalar multiple of the identity, and a nonzero scalar makes the normalized
transform a right inverse of the machine.  This is the article's endpoint with the parameter space
taken to be the intermediate coefficient space rather than parameter `L²`, which is the pair shape
the two bounds of this file can produce.

Nothing is assumed: the Schur step uses the irreducibility of the affine data representation, proved
in `LeanRidgelet.HA.AffineMackey`, and the representation on `Γ^k` is
`LeanRidgelet.quadraticSobolevContRepresentation`.  What remains for a concrete pair is to factor
the two Bochner integrals through the space, which is what
`LeanRidgelet.exists_quadraticCompositeIntertwiner_of_bounds` is about.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.quadratic_gamma_reconstruction_of_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (M : LeanRidgelet.QuadraticGammaMachine lam k)
  (R : LeanRidgelet.QuadraticGammaRidgelet lam k)
  (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume))
  (hf : M (R f) ≠ 0) :
  ∃ c,
    c ≠ 0 ∧
      LeanRidgelet.jointReconstructionOperator M R =
          c •
            ContinuousLinearMap.id ℂ
              ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) ∧
        Function.RightInverse ⇑(c⁻¹ • R.toContinuousLinearMap) ⇑M

theorem LeanRidgelet.quadratic_gamma_reconstruction_of_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] (k : ℕ)
  (M :
    LeanRidgelet.QuadraticGammaMachine lam
      k)
  (R :
    LeanRidgelet.QuadraticGammaRidgelet
      lam k)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume))
  (hf : M (R f) ≠ 0) :
  ∃ c,
    c ≠ 0 ∧
      LeanRidgelet.jointReconstructionOperator
            M R =
          c •
            ContinuousLinearMap.id ℂ
              ↥(MeasureTheory.Lp ℂ 2
                  MeasureTheory.volume) ∧
        Function.RightInverse
          ⇑(c⁻¹ • R.toContinuousLinearMap)
          ⇑M

**A nonzero constant through the intermediate space, from a probe.**  One datum whose image
under the composite is nonzero makes the constant nonzero, so the normalized transform inverts the
machine outright.  The reduction is the same as for the other shapes; only the parameter space
differs.

theoremdefined in LeanRidgelet/HA/QuadraticNonzero.lean

complete

theorem LeanRidgelet.quadraticComposite_intertwines_of_coeFn.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace (LeanRidgelet.QuadraticSymmetric E)]
  (lam : MeasureTheory.Measure (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (T :
    ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) →L[ℂ]
      ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume))
  (hT :
    ∀ (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)),
      ↑↑(T f) =ᵐ[MeasureTheory.volume]
        LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure lam)
          (LeanRidgelet.quadraticVectorFeature σ)
          (LeanRidgelet.bochnerRidgelet MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature ψ) ↑↑f))
  (g : E ≃ᵃ[ℝ] E) (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) :
  T
      (↑((LeanRidgelet.affineDataLpUnitaryRepresentation
              MeasureTheory.volume)
            g)
        f) =
    ↑((LeanRidgelet.affineDataLpUnitaryRepresentation
            MeasureTheory.volume)
          g)
      (T f)

theorem LeanRidgelet.quadraticComposite_intertwines_of_coeFn.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  [MeasurableSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  [BorelSpace
      (LeanRidgelet.QuadraticSymmetric E)]
  (lam :
    MeasureTheory.Measure
      (LeanRidgelet.QuadraticParameter E))
  [lam.IsAddHaarMeasure] {σ ψ : ℝ → ℂ}
  (T :
    ↥(MeasureTheory.Lp ℂ 2
          MeasureTheory.volume) →L[ℂ]
      ↥(MeasureTheory.Lp ℂ 2
          MeasureTheory.volume))
  (hT :
    ∀
      (f :
        ↥(MeasureTheory.Lp ℂ 2
            MeasureTheory.volume)),
      ↑↑(T f) =ᵐ[MeasureTheory.volume]
        LeanRidgelet.bochnerSynthesis
          (LeanRidgelet.quadraticRelativeMeasure
            lam)
          (LeanRidgelet.quadraticVectorFeature
            σ)
          (LeanRidgelet.bochnerRidgelet
            MeasureTheory.volume
            (LeanRidgelet.quadraticVectorFeature
              ψ)
            ↑↑f))
  (g : E ≃ᵃ[ℝ] E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)) :
  T
      (↑((LeanRidgelet.affineDataLpUnitaryRepresentation
              MeasureTheory.volume)
            g)
        f) =
    ↑((LeanRidgelet.affineDataLpUnitaryRepresentation
            MeasureTheory.volume)
          g)
      (T f)

**Any operator that realizes the composite pointwise intertwines the data representation.**  The
equivariance argument uses nothing about how the operator was built: only the two pointwise Bochner
intertwining identities of `LeanRidgelet.HA.QuadraticRelativeMeasure` and the almost-everywhere
formula for the data representation.  `LeanRidgelet.HA.QuadraticComposite` runs it for the operator
as a hypothesis, so it applies to an operator built any other way.

This is the shape of the article's universality claim, and it separates into a part that is proved and two analytic inputs that are not. What is proved: the reconstruction operator is its scalar times the identity; the pairing of a datum with its reconstruction is that scalar times the datum's squared norm, so the scalar is the pair's admissibility constant and not an artifact of the choice naming it; a single datum with nonzero image forces the scalar to be nonzero, hence gives the reconstruction outright, for a pair of intertwiners and for an endomorphism alike; and the composite estimate, which is the inequality the whole route exists to produce. None of that carries a placeholder.

The composite estimate deserves its statement read carefully. It says the composite of the two Bochner integrals is bounded on data L^2 by the product of the two constants — and it does not say that either factor is bounded on parameter L^2. The synthesis is bounded on \Gamma^k, which is smaller, the seminorm dominating the L^2 norm rather than the other way round. That asymmetry is the whole escape from the compactness obstruction, and it also fixes the shape of what remains: what the two bounds produce is a bounded intertwining endomorphism of data L^2, the article's weaker reading of its own boundedness hypothesis, not a bounded pair through parameter L^2. An earlier version of this development asked for the pair, which those bounds cannot give.

That endomorphism is now built, so the boundedness side carries no placeholder. Three things went into it: the composite estimate; a general tool that turns a pointwise formula into a bounded operator on L^2 from square integrability, almost-everywhere additivity and homogeneity, and a norm bound; and the observation that the equivariance argument uses nothing about how the operator was built, so it can be run with the pointwise realization taken as a hypothesis. The hypotheses beyond the two bounds are exactly the ones packaging requires and no estimate supplies: a formula linear on functions need not be linear on almost-everywhere classes, since splitting the defining integral over a sum needs each piece integrable.

One placeholder remains, and it is the admissibility statement — the article's own condition, and open mathematics rather than plumbing. Note that the nonzero constant of the adjoint route does not depend on it.

A nonzero constant is reached, by the coorbit route. Take the machine to be the adjoint of the ridgelet transform and the endpoint follows with no placeholder at all: one datum with nonzero transform gives a nonzero constant, the reconstruction operator is that constant times the identity, and the normalized ridgelet transform is a right inverse of the machine — the article's endpoint for Section 7, proved. The general statement and the reference are in the reconstruction chapter. What it costs is that the machine is the adjoint rather than an activation fixed in advance: at the level of the Bochner formulas that is the common-feature case, so this reconstructs with the network whose activation is the analysis feature. And the endpoint is available on the network's own integrals, not only on abstract operators: taking the synthesis feature to be the analysis feature makes the two Bochner integrals adjoint, so one datum with nonzero transform gives a nonzero constant for which the composite of the two integrals returns every datum multiplied by it. That is Section 7's claim written on the integrals the article writes down, with no placeholder — the hypotheses are the standing ones for analytic input, one Fubini condition, and the probe. The remaining placeholder is about the harder problem, an activation fixed in advance.

The endpoint is also available with the intermediate space as the parameter space: a machine out of \Gamma^k and a ridgelet transform into it compose to a scalar, and a probe makes the scalar nonzero. That is the pair shape the two bounds can produce — the synthesis is bounded on \Gamma^k, not on parameter L^2 — and it carries no placeholder either, the Schur step needing only the irreducibility of the data representation and the representation on the space being now bundled.

The admissibility statement is an existence over pairs deliberately. For a fixed pair it is false — the zero synthesis feature satisfies both bounds and annihilates every datum — so quantifying over all pairs satisfying the bounds would place a false statement behind a placeholder. The article fixes the activation to be the rectified linear unit, and specializing the existence to a fixed activation is the follow-up: it needs the activation's nonvanishing on the Fourier side together with the negative-order condition that controls its growth. Neither is assumed here, and no boundedness and no nonvanishing constant appear as hypotheses of the endpoint.