An extension of a theorem of Schoenberg to products of spheres

TL;DR

This paper extends Schoenberg's classical theorem to characterize continuous, isotropic, positive definite kernels on products of spheres, providing a broader understanding of such kernels in higher-dimensional settings.

Contribution

It generalizes Schoenberg's theorem from single spheres to products of spheres, offering a new characterization of positive definite kernels in this context.

Findings

Characterization of positive definite kernels on product spheres

Discussion of issues and future research directions

Extension of classical Schoenberg theorem

Abstract

We present a characterization for the continuous, isotropic and positive definite kernels on a product of spheres along the lines of a classical result of I. J. Schoenberg on positive definiteness on a single sphere. We also discuss a few issues regarding the characterization, including topics for future investigation.