On a new condition for strictly positive definite functions on spheres

Schreiner, Michael

doi:10.1090/S0002-9939-97-03634-4

On a new condition for strictly positive definite functions on spheres
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by Michael Schreiner

Proc. Amer. Math. Soc. 125 (1997), 531-539

DOI: https://doi.org/10.1090/S0002-9939-97-03634-4

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Abstract:

Recently, Xu and Cheney (1992) have proved that if all the Legendre coefficients of a zonal function defined on a sphere are positive then the function is strictly positive definite. It will be shown in this paper that, even if finitely many of the Legendre coefficients are zero, the strict positive definiteness can be assured. The results are based on approximation properties of singular integrals, and provide also a completely different proof of the results of Xu and Cheney.

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