9.1. Main results🔗

With a Gaussian input weight and a homogeneous Gaussian mixture on the direction parameter, the ridgelet transform on an infinite-dimensional Hilbert space keeps the structure of the finite-dimensional theory. The admissibility constant is a one-dimensional integral in which no dimension appears, the transform is injective with a Plancherel identity on \mathcal E_\alpha, and synthesis returns the target up to the frame operator T_\alpha, the correction induced by the weights, which is undone after synthesis, before analysis, or on the coefficients. Targets with a spectral density are integral networks with explicit coefficients; every continuous target is a compact-open limit of such networks; and sampling the coefficient gives finite-width networks at the rate N^{-1/2} when the density is regular along rays and the activation is globally Lipschitz. Weak reconstruction covers every non-polynomial tempered activation, and ReLU, tanh, the Gaussian distribution function and the Gaussian activation satisfy all of these conditions. The neural-operator layers of the examples are integral networks with explicit parameter measures to which the finite-width bounds apply.