AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-20
Constructive Universal Approximation by Infinite-Dimensional Neural Networks via Gaussian-Weighted Ridgelet Transform
Universal approximation results for neural networks with function inputs have mostly relied on existence arguments. This paper extends the ridgelet transform to continuous maps on infinite-dimensional Hilbert spaces, using Gaussian weights in place of the unavailable Lebesgue measure. The construction gives explicit coefficients and uniform approximation on compact sets, and finite-width networks achieve a dimension-free error of O(N^(-1/2)) in the width N. A companion Lean 4 formalization provides machine-checked proofs.