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 where $F$ is a finite abelian group. The characterisation obtained for these groups does not extend to arbitrary compact abelian groups; it fails in particular for all torsion-free groups.