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.
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Acknowledgements
We thank the reviewers for their helpful comments and suggestions. We are especially grateful to one reviewer for pointing out how to significantly simplify one of our proofs and for giving detailed and precise remarks on the improvement of the paper.
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Open Access funding enabled and organized by Projekt DEAL. The work of Janin Jäger was funded by the Deutsche Forschungsgemeinschaft (DFG - German research foundation) - Projektnummer: 461449252.
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Communicated by: Robert Schaback
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Hubbert, S., Jäger, J. Generalised Wendland functions for the sphere. Adv Comput Math 49, 3 (2023). https://doi.org/10.1007/s10444-022-10005-z
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DOI: https://doi.org/10.1007/s10444-022-10005-z