  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.
