$L^p$ Error Estimates for Approximation by Sobolev Splines and Wendland Functions on $\mathbb{R}^d$

TL;DR

This paper develops a new method to derive approximation error rates for Sobolev splines and Wendland functions, expanding understanding of their efficiency in multivariate approximation tasks.

Contribution

Introduces a convolution-based approach to obtain approximation rates for RBFs, including Sobolev splines and Wendland functions, using Green's function conditions and perturbation techniques.

Findings

Derived error rates for Sobolev splines as Green's functions.

Established approximation rates for Wendland functions.

Extended the theoretical framework for RBF approximation analysis.

Abstract

It is known that a Green's function-type condition may be used to derive rates for approximation by radial basis functions (RBFs). In this paper, we introduce a method for obtaining rates for approximation by functions which can be convolved with a finite Borel measure to form a Green's function. Following a description of the method, rates will be found for two classes of RBFs. Specifically, rates will be found for the Sobolev splines, which are Green's functions, and the perturbation technique will then be employed to determine rates for approximation by Wendland functions.