A simple remark on the order of approximation by compactly supported radial basis functions, and related networks

We consider simultaneous approximation of multivariate functions and their derivatives by using Wendland's compactly supported radial basis functions phi(s,k). By applying a greedy algorithm, it is presented that, regardless of dimension, an O(m(-1/2)) Order of approximation can be achieved by a linear combination of m translates of phi(s,k). A similar result on approximation by neural networks is established by using univariate radial functions as the activation functions of the networks.