Regular families of kernels for nonlinear approximation

Abstract

This article studies sufficient conditions on families of approximating kernels which provide N-term approximation errors from an associated nonlinear approximation space which match the best known orders of N-term wavelet expansion. These conditions provide a framework which encompasses some notable approximation kernels including splines, cardinal functions, and many radial basis functions such as the Gaussians and general multiquadrics. Examples of such kernels are given to justify the criteria. Additionally, the techniques involved allow for some new results on N-term Greedy interpolation of Sobolev functions via radial basis functions.