Derivatives of isotropic positive definite functions on spheres

Trübner, Mara; Ziegel, Johanna

doi:10.1090/proc/13561

Derivatives of isotropic positive definite functions on spheres
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by Mara Trübner and Johanna F. Ziegel

Proc. Amer. Math. Soc. 145 (2017), 3017-3031

DOI: https://doi.org/10.1090/proc/13561

Published electronically: January 25, 2017

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Abstract:

We show that isotropic positive definite functions on the $d$-dimensional sphere which are $2k$ times differentiable at zero have $2k+[(d-1)/2]$ continuous derivatives on $(0,\pi )$. This result is analogous to the result for radial positive definite functions on Euclidean spaces. We prove optimality of the result for all odd dimensions. The proof relies on montée, descente and turning bands operators on spheres which parallel the corresponding operators originating in the work of Matheron for radial positive definite functions on Euclidean spaces.

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