Strictly positive definite functions on compact abelian groups

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.