Characterization of Strict Positive Definiteness on products of complex spheres

TL;DR

This paper characterizes when functions on products of complex spheres are strictly positive definite, using conditions on their polynomial expansion coefficients, including special cases where parameters are 1 or infinity.

Contribution

It provides a necessary and sufficient condition for strict positive definiteness on complex sphere products based on polynomial expansion coefficients, extending to edge cases.

Findings

Derived a coefficient-based condition for strict positive definiteness.

Included analysis of cases with parameters 1 or infinity.

Established the intersection property with arithmetic progressions.

Abstract

In this paper we consider Positive Definite functions on products $\Omega_{2q}\times\Omega_{2p}$ of complex spheres, and we obtain a condition, in terms of the coefficients in their disc polynomial expansions, which is necessary and sufficient for the function to be Strictly Positive Definite. The result includes also the more delicate cases in which $p$ and/or $q$ can be $1$ or $\infty$. The condition we obtain states that a suitable set in $\mathbb{Z}^2$, containing the indexes of the strictly positive coefficients in the expansion, must intersect every product of arithmetic progressions.