Strictly positive definite functions on compact abelian groups

TL;DR

This paper characterizes strictly positive definite functions on certain compact abelian groups, specifically those of the form F × T^r, and highlights the limitations of this characterization for more general groups.

Contribution

It provides a Fourier-based characterization for strictly positive definite functions on groups of the form F × T^r, addressing a specific class of compact abelian groups.

Findings

Characterization holds for groups F × T^r, with F finite and r in natural numbers.

The characterization does not extend to all compact abelian groups, especially torsion-free groups.

The results clarify the structure of positive definite functions on these groups.

Abstract

We study the Fourier characterisation of strictly positive definite functions on compact abelian groups. Our main result settles the case $G = F \times \mathbb{T}^r$, with $r \in \mathbb{N}$ and $F$ finite. The characterisation obtained for these groups does not extend to arbitrary compact abelian groups; it fails in particular for all torsion-free groups.