Reproduction of Polynomials by Radial Basis Functions

For radial basis function interpolation of scattered data in IR d , the approximative reproduction of high-degree polynomials is studied. Results include uniform error bounds and convergence orders on compact sets. x1. Introduction We consider interpolation of real-valued functions f deened on a set IR d ; d 1. These functions are interpolated on a set X := fx 1 ; : : : ; x N X g of N X 1 pairwise distinct points x 1 ; : : : ; x N X in. Interpolation is done by linear combinations of translates (x ? x j) of a single continuous real-valued function deened on IR d. For various reasons it is sometimes necessary to add the space IP d m of d{variate polynomials of order not exceeding m to the interpolating functions. Interpolation is uniquely possible under the requirement If p 2 P d m satisses p(x i) = 0 for all x i 2 X then p = 0; (1) and if is conditionally positive deenite of order m (see e.g. 8]): Deenition 1. A function : IR d ! IR with (?x) = (x) is conditionally positive deenite of order m on IR d , if for all sets X = fx 1 X j=1 j p(x j) = 0 for all p 2 IP d m (2) the quadratic form P N X j;k=1 j k (x j ? x k) attains nonnegative values and vanishes only if = 0. We list a few examples where (x) := (kxk 2) is truly radial: