Strictly Positive Definite Functions on Compact Two-Point Homogeneous Spaces: the Product Alternative

TL;DR

This paper characterizes when the product of two isotropic positive definite kernels on compact two-point homogeneous spaces remains strictly positive definite, providing new methods for modeling in interpolation and approximation tasks.

Contribution

It offers necessary and sufficient conditions for the strict positive definiteness of kernel products on these spaces, extending to space-time settings with group actions.

Findings

Characterization of strict positive definiteness for kernel products

Conditions applicable to space-time kernel settings

New procedures for constructing valid interpolation models

Abstract

For two continuous and isotropic positive definite kernels on the same compact two-point homogeneous space, we determine necessary and sufficient conditions in order that their product be strictly positive definite. We also provide a similar characterization for kernels on the space-time setting $G \times S^d$, where $G$ is a locally compact group and $S^d$ is the unit sphere in $\mathbb{R}^{d+1}$, keeping isotropy of the kernels with respect to the $S^d$ component. Among other things, these results provide new procedures for the construction of valid models for interpolation and approximation on compact two-point homogeneous spaces.