Pointwise Estimates Formultivariate Interpolation Using Conditionally Positive Definite Functions

Abstract

. We seek pointwise error estimates for interpolants, on scattered data, constructed using a basis of conditionally positive definite functions of order m, and polynomials of degree not exceeding m-1. Two different approaches to the analysis of such interpolation are considered. The former uses distributions and reproducing kernel ideas, whilst the latter is based on a Lagrange function approach. Error estimates in terms of a point density measure are given for both methods of analysis. 1. Introduction Over recent years it has become clear that radial functions are very useful tools for multivariate approximation. Two radial functions in particular have found favour with practitioners, the multiquadric, h(x) = p 1 + kxk 2 , and the thin plate spline, h(x) = kxk 2 log kxk. In this paper we shall be considering the pointwise convergence of interpolation schemes employing conditionally positive definite (CPD) functions, a class which includes multiquadrics and thin plate splines, a..