Strictly positive definite kernels on a product of spheres

TL;DR

This paper characterizes strict positive definiteness of kernels on products of spheres, distinguishing between the usual and restricted cases, except when one sphere is a circle.

Contribution

It provides a comprehensive characterization of strict positive definiteness for kernels on product spheres, except for the circle case.

Findings

Characterization of strict positive definiteness on product spheres

Differentiation between usual and restricted cases

Remaining open case when one sphere is a circle

Abstract

For the real, continuous, isotropic and positive definite kernels on a product of spheres, one may consider not only its usual strict positive definiteness but also strict positive definiteness restrict to the points of the product that have distinct components. In this paper, we provide a characterization for strict positive definiteness in these two cases, settling all the cases but those in which one of the spheres is a circle.