Abstract
We develop a quantitative approximation theory for shallow neural networks using tools from time-frequency analysis. Working in weighted modulation spaces M^p,q_m(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^p,q_m(R^d) one can achieve f - f_N_W^n,r(Ω) N^-1/2f_M^p,q_m(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.