Native Hilbert Spaces for Radial Basis Functions I X1. Introduction and Overview

This contribution gives a partial survey over the native spaces associated to (not necessarily radial) basis functions. Starting from reproducing kernel Hilbert spaces and invariance properties, the general construction of native spaces is carried out for both the unconditionally and the conditionally positive deenite case. The deenitions of the latter are based on nitely supported functionals only. Fourier or other transforms are not required. The dependence of native spaces on the domain is studied, and criteria for functions and functionals to be in the native space are given. For the numerical treatment of functions of many variables, radial basis functions are useful tools. They have the form (kx ? yk 2) for vectors x; y 2 IR d with a univariate function deened on 0; 1) and the Euclidean norm k k 2 on IR d. This allows to work eeciently for large dimensions d, because the function boils the multivariate setting down to a univariate setting. Usually, the multivariate context comes back into play by picking a large number M of points x 1 ; : : :; x M in IR d and working with linear combinations s(x) := M X j=1 j (kx j ? xk 2): In certain cases, low{degree polynomials have to be added, but we give details later. Typical examples for radial functions (r) on r = kx ? yk 2 ; x; y 2 IR d ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.