Multivariate positive definite functions on spheres

TL;DR

This paper extends Schoenberg's theorem to multivariate Gegenbauer polynomials, providing new positive semidefinite constraints for spherical codes and advancing the understanding of positive definite functions on spheres.

Contribution

It generalizes Schoenberg's theorem to multivariate Gegenbauer polynomials, introducing novel constraints for spherical code analysis.

Findings

Derived new positive semidefinite constraints for spherical codes

Extended Schoenberg's theorem to multivariate Gegenbauer polynomials

Enhanced tools for analyzing positive definite functions on spheres

Abstract

In 1942 I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a positive linear combination of the Gegenbauer polynomials. In this paper we extend Schoenberg's theorem for multivariate Gegenbauer polynomials. This extension derives new positive semidefinite constraints for the distance distribution which can be applied for spherical codes.