Strict positive definiteness of convolutional and axially symmetric kernels on d-dimensional spheres

TL;DR

This paper establishes new sufficient conditions for strict positive definiteness of convolutional and axially symmetric kernels on spheres, expanding previous results to more general kernel structures using spherical harmonics.

Contribution

It provides the first comprehensive criteria for strict positive definiteness of non-radial kernels on spheres with specific coefficient structures, generalizing prior radial kernel results.

Findings

Generalized positive definiteness conditions for convolutional kernels

Extended criteria to axially symmetric kernels on spheres

Unified framework using spherical harmonics

Abstract

The paper introduces new sufficient conditions of strict positive definiteness for kernels on d-dimensional spheres which are not radially symmetric but possess specific coefficient structures. The results use the series expansion of the kernel in spherical harmonics. The kernels either have a convolutional form or are axially symmetric with respect to one axis. The given results on convolutional kernels generalise the result derived by Chen et al. [8] for radial kernels. Keywords: strictly positive definite kernels, covariance functions, sphere