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 $\mathrm{ReLU}^k$ activations.

The central object is the Radon-domain $L^p$ space $\mathcal{R}L^p_k(\Omega)$ containing all functions on a bounded domain $\Omega$ 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 $H^{k+(d+1)/2}(\Omega)$.

For general $1<p<\infty$, 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 $O\!\big(n^{-\frac12-\frac{2k+1}{2d}}\big)$ at $p=2$ for linearized neural networks.