  This paper considers the problem of restricting the short-time Fourier
transform to domains of nonzero measure in the plane and studies sampling
bounds of such systems. In particular, we give a quantitative estimate for the
lower sampling bound in the case of Hermite windows and derive a sufficient
condition for a large class of windows in terms of a certain planar density. On
the way, we prove a Remez-type inequality for polyanalytic functions. MSC2010:
42C40, 46E15, 46E20, 42C15
