Generalised Wendland functions for the sphere

Abstract

AbstractIn this paper, we compute the spherical Fourier expansion coefficients for the restriction of the generalised Wendland functions fromd-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 certain3F2hypergeometric function. We present tight asymptotic bounds on the decay rate of the spherical Fourier coefficients and, in the case wheredis odd, we are able to provide the precise asymptotic rate of decay. Numerical evidence suggests that this precise asymptotic rate also holds whendis 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.