Sobolev-type Error Estimates for Interpolation by Radial Basis Functions

We generalize techniques dating back to Duchon 4] for error estimates for interpolation by thin plate splines to basis functions with positive and algebraically decaying Fourier transform. We include L p-estimates for 1 p < 2 that can also be applied to thin plate spline approximation. x1. Introduction Radial basis functions are a well-established tool for multivariate approximation problems. A radial basis interpolant to a continuous function f : IR d ! IR on a set X = fx 1 ; : : : ; x N g is formed by s f (x) = N X j=1 j (x ? x j): Here : IR d ! IR is a xed, positive deenite and symmetric function, and the coeecients j are determined by the interpolation conditions s f (x j) = f(x j), 1 j N. A more general setting adds certain polynomials to s f to form the interpolant and allows to be a more general function. For details we refer the reader to the overview articles 3, 5, 6, 8]. In many cases, the function is radial in the sense (x) = (kxk 2), x 2 IR d. In this paper we are mainly interested in basis functions : IR d ! IR that are in L 1 (IR d) and possess Fourier transforms ^ (!) = (2) ?d=2 Z I R d (x)e ix T ! dx which satisfy c 1 (1 + k!k 2) ?d?2k?1 ^ (!) c 2 (1 + k!k 2) ?d?2k?1 ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.