Convolution roots and differentiability of isotropic positive definite functions on spheres
Probability
2013-10-29 v3 Statistics Theory
Statistics Theory
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.
Keywords
Cite
@article{arxiv.1201.5833,
title = {Convolution roots and differentiability of isotropic positive definite functions on spheres},
author = {Johanna Ziegel},
journal= {arXiv preprint arXiv:1201.5833},
year = {2013}
}