Strictly positive definite non-isotropic kernels on two-point homogeneous manifolds: The asymptotic approach

TL;DR

This paper establishes conditions under which certain positive definite kernels on symmetric manifolds are strictly positive definite, extending previous isotropic kernel results and analyzing the real sphere case in detail.

Contribution

It provides a new asymptotic approach to characterize strict positive definiteness of non-isotropic kernels on two-point homogeneous manifolds.

Findings

Derived sufficient conditions for strict positive definiteness

Generalized isotropic kernels to non-isotropic cases

Detailed analysis of the real sphere case

Abstract

We present sufficient condition for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their representation as a series of spherical harmonics. The family analyzed is a generalization of the isotropic kernels and the case of a real sphere is analyzed in details.