Relations between Schoenberg Coefficients on Real and Complex Spheres of Different Dimensions

TL;DR

This paper explores the relationships between Schoenberg coefficients on real and complex spheres across various dimensions, providing new insights into their spectral properties and applications to positive definiteness.

Contribution

It introduces novel relations between Schoenberg sequences on real and complex spheres of different dimensions, enhancing understanding of spectral representations.

Findings

Derived relations between Schoenberg sequences on real and complex spheres

Applied findings to characterize strict positive definiteness

Enhanced spectral analysis of positive definite functions

Abstract

Positive definite functions on spheres have received an increasing interest in many branches of mathematics and statistics. In particular, the Schoenberg sequences in the spectral representation of positive definite functions have been studied by several mathematicians in the last years. This paper provides a set of relations between Schoenberg sequences defined over real as well as complex spheres of different dimensions. We illustrate our findings describing an application to strict positive definiteness.