Convolution roots and differentiability of isotropic positive definite functions on spheres

Ziegel, Johanna

doi:10.1090/S0002-9939-2014-11989-7

Convolution roots and differentiability of isotropic positive definite functions on spheres
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by Johanna Ziegel

Proc. Amer. Math. Soc. 142 (2014), 2063-2077

DOI: https://doi.org/10.1090/S0002-9939-2014-11989-7

Published electronically: February 28, 2014

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Abstract:

We prove that any isotropic positive definite function on the sphere can be written as the spherical self-convolution of an isotropic real-valued function. It is known that isotropic positive definite functions on $d$-dimensional Euclidean space admit a continuous derivative of order $[(d-1)/2]$. We show that the same holds true for isotropic positive definite functions on spheres and prove that this result is optimal for all odd dimensions.

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