Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation

We construct shallow neural networks with fixed Gaussian and tanh-difference activations on bounded Lipschitz domains $Ω\subset\mathbb{R}^d$, $d\ge2$. For each $s>0$, explicit deterministic rules prescribe all hidden parameters independently of the target $u\in H^s(Ω)$. At frequency resolution $M$,