Comparison of Radial Basis Function Interpolants

Abstract

This paper compares radial basis function interpolants on different spaces. The spaces are generated by other radial basis functions, and comparison is done via an explicit representation of the norm of the error functional. The results pose some new questions for further research. x1. Introduction We consider interpolation of real-valued functions f defined on a set \\Omega ` IR d ; d 1. These functions are evaluated on a set X := fx 1 ; : : : ; xNX g of NX 1 pairwise distinct points x 1 ; : : : ; xNX in \\Omega\\Gamma If N 2; d 2 and\\Omega ` IR d are given with\\Omega containing at least an interior point, it is well known that there is no N-dimensional space of continuous functions on\\Omega that contains a unique interpolant for every f and every set X = fx 1 ; : : : ; xNX g ae\\Omega ` IR d consisting of N = NX data points. Thus the family of interpolants must necessarily depend on X. This can easily be achieved by using translates \\Phi(x \\Gamma x j ) of a single continu...