  We prove the sharp embedding between the spectral Barron space and the Besov
space. Given the spectral Barron space as the target function space, we prove a
dimension-free 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 $s$ is the smoothness index of the spectral Barron space.
