Radial basis functions and corresponding zonal series expansions on the sphere

Abstract

Since radial positive definite functions on R d remain positive definite when restricted to the sphere, it is natural to ask for properties of the zonal series expansion of such functions which relate to properties of the Fourier–Bessel transform of the radial function. We show that the decay of the Gegenbauer coefficients is determined by the behavior of the Fourier–Bessel transform at the origin.