Extending the Range of Error Estimates for Radial Approximation in Euclidean Space and on Spheres

R A Brownlee,E H Georgoulis,J Levesley

doi:10.1137/060650428

Abstract

We adapt Schaback's error doubling trick [R. Schaback, Math. Comp., 68 (1999), pp. 201–216] to give error estimates for radial interpolation of functions with smoothness lying (in some sense) between that of the usual native space and the subspace with double the smoothness. We do this for both bounded subsets of ${\mathbb{R}}^d$ and spheres. As a step on the way to our ultimate goal we also show convergence of pseudoderivatives of the interpolation error.

Highlights

In this paper we are interested in extending the range of applicability of error estimates for radial basis function interpolation in Euclidean space and on spheres
Given a univariate function φ defined either on R+ or [0, π], depending on whether we are in Euclidean space or on the sphere, we form an approximation
Y ∈ Y, are determined by the interpolation conditions SφY (y) = f (y), for y ∈ Y, we refer to SφY as the φ-spline interpolant to f on Y

Summary

Introduction

In this paper we are interested in extending the range of applicability of error estimates for radial basis function interpolation in Euclidean space and on spheres. Given a univariate function φ defined either on R+ or [0, π], depending on whether we are in Euclidean space or on the sphere, we form an approximation The usual error estimate for φ-spline interpolants is of the form

Results

Conclusion