2 - Approximation and interpolation with radial functions

Summary

Abstract

This chapter gives a short, up-to-date survey of some recent developments in the research on radial basis functions. Among its other new achievements, we consider results on convergence rates of interpolation with radial basis functions, and also recent contributions on approximation on spheres and on computation of interpolants with Krylov space methods.

Introduction

Research into radial basis functions is an immensely active and fruitful field at present and it is important and worthwhile to stand back and summarize the newest developments from time to time. In brief, this is the goal of this chapter, although we will by necessity be far from comprehensive. One of the most important aspects from the perspective of approximation theorists is the accuracy of approximation with radial basis functions when the centers are scattered. This is a subject quite suitable to begin this review with, as the whole development of radial basis functions was initiated by Duchon's contributions (1976,1978,1979) on exactly this question in a special context, especially for thin-plate splines approximation in ℝ².

Before we begin, we recall what is understood by approximation and interpolation by radial basis function. We always start with a univariate continuous function – the radial function – φ that is radialized by composition with the Euclidean norm on ℝⁿ, or a suitable replacement thereof when we are working on an (n − 1) sphere in n-dimensional Euclidean space.