Generalised Wendland functions for the sphere

Hubbert, Simon and Janin, J. (2023) Generalised Wendland functions for the sphere. Advances in Computational Mathematics 49 (3), ISSN 1019-7168.

Abstract

In this paper we compute the spherical Fourier expansions 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 $_{3}F_{2}$ 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.

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