Spectral Barron space for deep neural network approximation

We prove the sharp embedding between the spectral Barron space and the Besov space with embedding constants independent of the input dimension. Given the spectral Barron space as the target function space, we prove a dimension-free convergence result that if the neural network contains $L$ hidden layers with $N$ units per layer, then the upper and lower bounds of the $L^2$-approximation error are $\mathcal{O}(N^{-sL})$ with $0 < sL\le 1/2$, where $s\ge 0$ is the smoothness index of the spectral Barron space.