  We study the approximation capacity of some variation spaces corresponding to
shallow ReLU$^k$ neural networks. It is shown that sufficiently smooth
functions are contained in these spaces with finite variation norms. For
functions with less smoothness, the approximation rates in terms of the
variation norm are established. Using these results, we are able to prove the
optimal approximation rates in terms of the number of neurons for shallow
ReLU$^k$ neural networks. It is also shown how these results can be used to
derive approximation bounds for deep neural networks and convolutional neural
networks (CNNs). As applications, we study convergence rates for nonparametric
regression using three ReLU neural network models: shallow neural network,
over-parameterized neural network, and CNN. In particular, we show that shallow
neural networks can achieve the minimax optimal rates for learning H\"older
functions, which complements recent results for deep neural networks. It is
also proven that over-parameterized (deep or shallow) neural networks can
achieve nearly optimal rates for nonparametric regression.
