Radial basis functions and corresponding zonal series expansions on the sphere

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. (c) 2005 Elsevier Inc. All rights reserved.