Strictly positive definite functions on compact abelian 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.

9 pages; submitted to Proc. AMS