Generalised Wendland functions for the sphere

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. The development required to derive these coefficients relies heavily upon known asymptotic results for hypergeometric functions and the final result shows that they can be expressed in closed form as a multiple of a certain ${3}F{2}$ hypergeometric function. Using the closed form expressions we are able to provide the precise asymptotic rates of decay for the spherical Fourier coefficients which we observe have a close connection to the asymptotic decay rate of the corresponding Euclidean Fourier transform.