Error Estimates for Approximation by Radial Basis Functions

Abstract

This paper considers the problem of error estimates for interpolation by radial basis functions. To this end, a recap of the theory of bounding linear functionals in Hilbert spaces is presented. We begin with a normed linear space X. Let γ ₁,…, γ _m be linear ‘information functionals’ on X such that, for a given f ∈ X, the ‘information’ γ _i (f) = α _i , i = 1,…, m is known. With this data, we wish to compute γ(f) where γ is another linear functional on X. By taking a set of distinct points A = {a ₁,…, a _m } and choosing the information functionals to be point evaluations, that is, γ _i (f) = f(a _i ), i = 1,…, m with γ(f) = f(x) for some fixed point x, we obtain a general interpolation problem. We will, however, concentrate on interpolation by radial basis functions and it will become clear that the analysis which leads to the error estimates can subsequently be used to characterise the interpolant itself. Thus, the theory presented here is a very powerful technique in interpolation theory.