Isotropic Positive Definite Functions on Spheres

TL;DR

This paper explores the relationship between positive definite functions on spheres and Euclidean spaces, introducing new techniques for odd dimensions and verifying conjectures about positive definiteness.

Contribution

Develops a new method for relating positive definite functions on spheres and Euclidean spaces, and proves a specific class of functions are positive definite on spheres.

Findings

Established inheritance of positive definiteness between $ r^d$ and $S^d$ for odd $d$

Proved a class of functions are positive definite on $S^2$

Verified a conjecture and a Pólya type criterion for spheres.

Abstract

In this paper, we investigate the relationship between positive definite functions on the unit sphere $\sph$ and on the Euclidean space $\RR^d$. For the dimension $d$ to be odd, a new technique is developed to establish the inheritance of positive (semi-)definite property from $\RR^d$ to $\sph$ and the converse. For $d=2$, it is proved that a function defined by $$f_{\t,\delta}(t)=(\t-t)_+^\delta, \quad \delta\geq \f{d+1}2 $$ is positive definite on the unit sphere $\mathbb{S}^2$ by restricting $\t$ in an absolute range. Our results can verify a conjecture proposed by R.K. Beatson, W. zu Castell, Y. Xu and a sharp P\'{o}lya type criterion for positive definite functions on spheres.