Wendland functions with increasing smoothness converge to a Gaussian

TL;DR

This paper proves that Wendland functions, a class of radial basis functions, converge uniformly to a Gaussian as their smoothness parameter increases, supported by numerical experiments.

Contribution

It demonstrates that Wendland functions with increasing smoothness converge to a Gaussian, providing a new understanding of their limiting behavior.

Findings

Wendland functions converge to a Gaussian as smoothness increases

Numerical experiments support the theoretical convergence

The convergence is uniform with a linear change of variables

Abstract

The Wendland functions are a class of compactly supported radial basis functions with a user-specified smoothness parameter. We prove that with a linear change of variables, both the original and the "missing" Wendland functions converge uniformly to a Gaussian as the smoothness parameter approaches infinity. We also give numerical experiments with Wendland functions of different smoothness.