Generalised Wendland functions for the sphere

Abstract

In this paper, we compute the spherical Fourier expansion coefficients for the restriction of the generalised Wendland functions from d-dimensional Euclidean space to the (d − 1)-dimensional unit sphere. We use results from the theory of special functions to show that they can be expressed in a closed form as a multiple of a certain 3F2 hypergeometric function. We present tight asymptotic bounds on the decay rate of the spherical Fourier coefficients and, in the case where d is odd, we are able to provide the precise asymptotic rate of decay. Numerical evidence suggests that this precise asymptotic rate also holds when d is even and we pose this as an open problem. Finally, we observe a close connection between the asymptotic decay rate of the spherical Fourier coefficients and that of the corresponding Euclidean Fourier transform.