Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain $L^p$ Ridge Integral Spaces for ReLU${}^k$ Networks

We develop the $L^p$ space and approximation theory for shallow neural networks with ${\mathrm{R}\mathrm{e}\mathrm{L}\mathrm{U}}^k$ activations. The central object is the Radon-domain $L^p$ space $\mathcal{R}{L}_k^p(\mathrm{\Omega })$ containing all functions on a bounded domain $\mathrm{\Omega }$ that admit a ridge integral representation whose coefficient density belongs to $L^p$ in the Radon domain. In the Hilbert case $p=2$, we prove by elementary Fourier analysis that this space recovers the critical Sobolev space $H^{k+(d+1)\mathrm{/}2}(\mathrm{\Omega })$. For general $1<p<\mathrm{\infty }$, the identity becomes a Sobolev sandwich. The sharp gap of each side is exactly the Seeger–Sogge–Stein loss for the Radon transform as a Fourier integral operator. This also clarifies how the activation regularity and Radon back-projection jointly produce the regularity. As an application, we discretize the integral representation using a deterministic interpolation skeleton plus uniform sampling. This yields high-probability $L^p$ approximation rates and the optimal Hilbert rate $O\text{\hspace{0.17em}⁣}\left(n^{-\frac{1}{2}-\frac{2k+1}{2d}}\right)$ at $p=2$ for linearized neural networks.