Infinite-dimensional operator ridgelet transform

1.2. Main results🔗

The universality statement is of a type already known; the novelty lies in how the approximants are obtained. The assumptions on filters and activations belong to the individual statements.

  • Theorem 3.14 proves the Plancherel identity, closed range and injectivity on the Hilbert space \mathcal E_\alpha.

  • Theorem 4.5 gives explicit coefficients for targets with a spectral density, including an absolutely convergent network integral under regularity along rays.

  • Theorem 4.8 identifies the frame operator as the Riesz isomorphism and gives reconstruction, backprojection and inversion formulas.

  • Theorem 6.6 proves finite total variation and parameter moments from regularity along rays, connecting representation to finite-width approximation.

  • Theorem 6.8 gives constructive compact-open universality through spectral targets, with approximation rates for the resulting integral networks.

The frame operator T_\alpha is the correction induced by the weights. Analysis followed by synthesis gives C T_\alpha f; applying C^{-1}T_\alpha^{-1} recovers f. The correction can equivalently be made before analysis or on the coefficients. Different admissible analysis filters give different coefficients for the same target, so the representation is a family indexed by the filter.

Scope. The coefficient formula is explicit once a spectral density is given; for an arbitrary continuous target, the universality proof first chooses a spectral approximation, and the variation and moment constants depend on that choice. No uniform width–accuracy relation for all continuous targets, coefficient-computation algorithm or statistical learning guarantee follows. Universality is proved in the compact-open topology, not globally on weighted non-compact spaces, and the N^{-1/2} rate is a Barron-type statement for targets whose coefficient has finite variation and moments, consistent with the known lower bounds for Lipschitz operator classes. Every finite-width network factors through finitely many input measurements; representing the directions themselves with finitely many numbers is a further approximation problem. Reconstruction with a tempered distribution as the activation takes place in a dual space, and general nonlinear operator activations are outside that reconstruction theorem.