Convolution roots and differentiability of isotropic positive definite functions on spheres

TL;DR

This paper demonstrates that isotropic positive definite functions on spheres can be expressed as spherical self-convolutions and possess optimal differentiability properties similar to Euclidean spaces.

Contribution

It establishes the convolution representation for these functions on spheres and proves their differentiability order is optimal in odd dimensions.

Findings

Isotropic positive definite functions on spheres can be written as spherical self-convolutions.

Such functions have a continuous derivative of order [(d-1)/2], matching Euclidean space results.

Differentiability order is proven to be optimal for all odd dimensions.

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.