Convolution roots and differentiability of isotropic positive definite functions on spheres

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.

References in corpus (2)

• Strictly and non-strictly positive definite functions on spheres
• A Pólya criterion for (strict) positive definiteness on the sphere

Cited by in corpus (1)

• From Fourier to Gegenbauer: Dimension walks on spheres