  In this paper we investigate new results on the theory of superoscillations
using time-frequency analysis tools and techniques such as the short-time
Fourier transform (STFT) and the Zak transform. We start by studying how the
short-time Fourier transform acts on superoscillation sequences. We then apply
the supershift property to prove that the short-time Fourier transform
preserves the superoscillatory behavior by taking the limit. It turns out that
these computations lead to interesting connections with various features of
time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames,
2D-complex Hermite polynomials, and polyanalytic functions. We treat different
cases depending on the choice of the window function moving from the general
case to more specific cases involving the Gaussian and the Hermite windows. We
consider also an evolution problem with an initial datum given by
superoscillation multiplied by the time-frequency shifts of a generic window
function. Finally, we compute the action of STFT on the approximating sequences
with a given Hermite window.
