1.1. From function inputs to network coefficients🔗

Operator learning approximates nonlinear maps whose inputs, and sometimes outputs, are functions. A basic model on a real Hilbert space H, with outputs in Y, is f_N(x)=\sum_{j=1}^N v_j\sigma(\langle a_j,x\rangle-b_j). Each direction a_j is a linear measurement of the input. The question here is how to choose the directions, biases and output coefficients from a target function.

An integral representation replaces the finite sum by f(x)=S_\sigma[\Gamma](x)=\int_{H\times\mathbb R}\sigma(\langle a,x\rangle-b)\,\Gamma(\mathrm da,\mathrm db). The finite scalar or vector measure \Gamma encodes the representation. Its normalized variation is a probability law for sampling neurons, and its polar density supplies their output weights. Variation and parameter moments then control the approximation error.

A ridgelet transform sends a target to coefficients indexed by directions and biases. The construction below is made directly on H; it does not begin by selecting a finite-dimensional input encoder. Universality of such networks is usually proved by existence arguments, by reduction to a finite-dimensional theorem or by Stone–Weierstrass-type density on the infinite-dimensional space, and Hilbert-input random-feature rates are known; those results establish existence or sampling bounds without an analysis operator. The construction here supplies that operator: an explicit coefficient formula, a norm identity and reconstruction, for the full class of continuous non-polynomial activations of polynomial growth, from which universality and finite-width rates follow.

The main technical obstacle is that an infinite-dimensional Hilbert space has no locally finite translation-invariant analogue of Lebesgue measure. We integrate against a Gaussian input measure and a homogeneous mixture of Gaussian direction measures; the homogeneity parameter of the mixture replaces the dimension-dependent factor of the finite-dimensional admissibility condition. The measure construction and Fourier identities are given in Section 3, before the reconstruction and approximation theorems use them.