Strictly Positive Definite Kernels on a Product of Spheres II

TL;DR

This paper establishes a necessary and sufficient condition for the strict positive definiteness of isotropic kernels on the product of a circle and a higher dimensional sphere, extending previous results on products of circles and spheres.

Contribution

It provides a new characterization of strict positive definiteness for kernels on mixed-dimensional sphere products, filling a gap in the existing theory.

Findings

Derived a necessary and sufficient condition for strict positive definiteness.

Extended previous results from products of circles and high-dimensional spheres.

Complemented existing literature with new theoretical insights.

Abstract

We present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result complements similar results previously obtained for strict positive definiteness on a product of circles [Positivity, to appear, arXiv:1505.01169] and on a product of high dimensional spheres [J. Math. Anal. Appl. 435 (2016), 286-301, arXiv:1505.03695].