  In this paper, we provide a strategy to determine the eigenvalue decay rate
(EDR) of a large class of kernel functions defined on a general domain rather
than $\mathbb S^{d}$. This class of kernel functions include but are not
limited to the neural tangent kernel associated with neural networks with
different depths and various activation functions. After proving that the
dynamics of training the wide neural networks uniformly approximated that of
the neural tangent kernel regression on general domains, we can further
illustrate the minimax optimality of the wide neural network provided that the
underground truth function $f\in [\mathcal H_{\mathrm{NTK}}]^{s}$, an
interpolation space associated with the RKHS $\mathcal{H}_{\mathrm{NTK}}$ of
NTK. We also showed that the overfitted neural network can not generalize well.
We believe our approach for determining the EDR of kernels might be also of
independent interests.
