Time-frequency analysis for neural networks

We develop a quantitative approximation theory for shallow neural networks using tools from time-frequency analysis. Working in weighted modulation spaces M m p , q ( R d ) , we prove dimension-independent approximation rates in Sobolev norms W n,r (Ω) for networks whose units combine standard activations with localized time-frequency windows. Our main result shows that for f ∈ M m p , q ( R d ) one can achieve ∥ f − f N ∥ W n , r ( Ω ) ≲ N − 1 / 2 ∥ f ∥ M m p , q ( R d ) , on bounded domains, with explicit control of all constants. We further obtain global approximation theorems on R d using weighted modulation dictionaries, and derive consequences for Feichtinger’s algebra, Fourier-Lebesgue spaces, and Barron spaces. Numerical experiments in one and two dimensions confirm that modulation-based networks achieve substantially better Sobolev approximation than standard ReLU networks, consistent with the theoretical estimates.