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 $\mathcal{R}L^p_k(Ω)$ containing all functions on a bounded domain $Ω$ 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 $H^{k+(d+1)/2}(Ω)$. For general $1<p...