$L^p$ sampling numbers for the Fourier-analytic Barron space

In this paper, we consider Barron functions $f:[0,1{]}^d\to \mathbb{R}$ of smoothness $\sigma >0$, which are functions that can be written as $$f(x)={\int }_{{\mathbb{R}}^d}F(\xi )e^{2\pi i⟨x,\xi ⟩}d\xi \text{with}{\int }_{{\mathbb{R}}^d}\mathrm{\mid }F(\xi )\mathrm{\mid }\cdot (1+\mathrm{\mid }\xi \mathrm{\mid }{)}^{\sigma }d\xi <\mathrm{\infty }\mathrm{.}$$ For $\sigma =1$, these functions play a prominent role in machine learning, since they can be efficiently approximated by (shallow) neural networks without suffering from the curse of dimensionality. For these functions, we study the following question: Given $m$ point samples $f(x_1),\dots ,f(x_m)$ of an unknown Barron function $f:[0,1{]}^d\to \mathbb{R}$ of smoothness $\sigma$, how well can $f$ be recovered from these samples, for an optimal choice of the sampling points and the reconstruction procedure? Denoting the optimal reconstruction error measured in $L^p$ by $s_m(\sigma ;L^p)$, we show that $$m^{-\frac{1}{\max \{p,2\}}-\frac{\sigma }{d}}\lesssim s_m(\sigma ;L^p)\lesssim (\ln (e+m){)}^{\alpha (\sigma ,d)\mathrm{/}p}\cdot m^{-\frac{1}{\max \{p,2\}}-\frac{\sigma }{d}},$$ where the implied constants only depend on $\sigma$ and $d$ and where $\alpha (\sigma ,d)$ stays bounded as $d\to \mathrm{\infty }$.