Generalised Wendland functions for the sphere

TL;DR

This paper derives explicit formulas for spherical Fourier coefficients of generalized Wendland functions restricted to the sphere, revealing their decay rates and connections to Euclidean Fourier transforms.

Contribution

It provides the first closed-form expressions for these coefficients and analyzes their asymptotic decay, linking spherical and Euclidean Fourier analysis.

Findings

Closed-form expressions for spherical Fourier coefficients

Asymptotic decay rates of coefficients determined

Connection established between spherical and Euclidean Fourier transforms

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.