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 Mp,qₘ(Rᵈ), we prove dimension-independent approximation rates in Sobolev norms Wn,r(Ω) for networks whose units combine standard activations with localized time-frequency windows. Our main result shows that for f ∈ Mp,qₘ(Rᵈ) one can achieve \[ \|f - f_N\|_{Wn,r(Ω)} N-1/2\,\|f\|_{Mp,q_m(Rᵈ)}, \] on bounded domains, with explicit control of all constants. We further obtain global approximation theorems on Rᵈ 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.