  A careful study of the physical properties of a family of coherent states on
the circle, introduced some years ago by de Bi\`evre and Gonz\'alez in [DG 92],
is carried out. They were obtained from the Weyl-Heisenberg coherent states in
$L^2(\R)$ by means of the Weil-Brezin-Zak transformation, they are labeled by
the points of the cylinder $S^1 \times \R$, and they provide a realization of
$L^2(S^1)$ by entire functions (similar to the well-known Fock-Bargmann
construction). In particular, we compute the expectation values of the position
and momentum operators on the circle and we discuss the Heisenberg uncertainty
relation.
