A unified theory of radial basis functions: Native Hilbert spaces for radial basis functions II

This contribution provides a new formulation of the theory of radial basis functions in the context of integral operators. Instead of Fourier transforms, the most important tools now are expansions into eigenfunctions. This unifies the theory of radial basis functions in Rd with the theory of zonal functions on the sphere Sd−1 and the theory of kernel functions on Riemannian manifolds. New characterizations of native spaces and positive definite functions are included. The paper is a self-contained continuation of an earlier survey (R. Schaback, International Series of Numerical Mathematics, Vol. 132, Birkhäuser, Basel, 1999, pp. 255–282) over the native spaces associated to (not necessarily radial) basis functions.