  We obtain estimates for the $L^{p}$-norm of the short-time Fourier transform
(STFT) for functions in modulation spaces, providing information about the
concentration on a given subset of $\mathbb{R}^{2}$, leading to deterministic
guarantees for perfect reconstruction using convex optimization methods. More
precisely, we will obtain large sieve inequalities of the Donoho-Logan type,
but instead of localizing the signals in regions $T\times W$ of the
time-frequency plane using the Fourier transform to intertwine time and
frequency, we will localize the representation of the signals in terms of the
short-time Fourier transform in sets $\Delta $ with arbitrary geometry. At the
technical level, since there is no proper analogue of Beurling's extremal
function in the STFT setting, we introduce a new method, which rests on a
combination of an argument similar to Schur's test with an extension of Seip's
local reproducing formula to general Hermite windows. When the windows are
Hermite functions, we obtain local reproducing formulas for polyanalytic Fock
spaces which lead to explicit large sieve constant estimates and, as a
byproduct, to a reconstruction formula for $f\in L^{2}(\mathbb{R})$ from its
STFT values on arbitrary discs. A discussion on optimality follows, along the
lines of Donoho-Stark paper on uncertainty principles and signal recovery. We
also consider the case of discrete Gabor systems, vector-valued STFT transforms
and rephrase the results in terms of the polyanalytic Bargmann-Fock transforms.
