Derivatives of isotropic positive definite functions on spheres

TL;DR

This paper establishes the smoothness properties of isotropic positive definite functions on spheres, showing how differentiability at zero relates to the number of continuous derivatives on the sphere, with proofs of optimality in odd dimensions.

Contribution

It extends the understanding of smoothness of isotropic positive definite functions on spheres, paralleling Euclidean results, and introduces operators that facilitate these proofs.

Findings

Differentiability at zero implies specific smoothness on the sphere.

Optimality of the smoothness result is proven for all odd dimensions.

Operators analogous to Euclidean space are used to analyze functions on spheres.

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 mont\'ee, descente and turning bands operators on spheres which parallel the corresponding operators originating in the work of Matheron for radial positive definite functions on Euclidian spaces.