We develop an alternative approach to the study of Fourier series, based on the Short-Time-Fourier Transform (STFT) acting on \(L_{\nu }^{2}(0,1)\) , the space of measurable functions f in \(\mathbb {R}\) , square-integrable in (0, 1), and time-periodic up to a phase factor: for fixed \(\nu \in \mathbb {R}\) , \(\begin{aligned} f(t+k)=e^{2\pi ik\nu }f(t){, \ }k\in \mathbb {Z}\text {.} \end{aligned}\) The resulting phase space is the vertical strip \(\mathbb {C}/\mathbb {Z}=[0,1)\times \mathbb {R}\) , a flat model of an infinite cylinder, which leads to Gabor frames with an interesting structure theory, allowing for a Janssen-type representation. As expected, a Gaussian window leads to a Fock space of entire functions, studied in the companion paper by the same authors [Beurling-type density theorems for sampling and interpolation on the flat cylinder]. When g is a Hermite function, we are lead to true Fock spaces of polyanalytic functions (Landau level eigenspaces) on the vertical strip \([0,1)\times \mathbb {R}\) . We first prove a density condition for a lattice to be interpolating in this space. Furthermore, an analogue of the sufficient Wexler-Raz conditions is obtained which leads to new criteria for Gabor frames in \(L^{2}(\mathbb {R})\) , and to sufficient conditions for Gabor frames in \(L_{\nu }^{2}(0,1)\) with Hermite windows (an analogue of a theorem of Gröchenig and Lyubarskii about Gabor frames with Hermite windows) and with totally positive windows in the Feichtinger algebra (an analogue of a recent theorem of Gröchenig). We also consider a vectorial STFT in \(L_{\nu }^{2}(0,1)\) and, using the vector with the first Hermite functions as window, we introduce the (full) Fock spaces of polyanalytic functions on \([0,1)\times \mathbb {R}\) and their associated Bargmann-type transforms, and prove an analogue of Vasilevski’s orthogonal decomposition into true polyanalytic Fock spaces (Landau level eigenspaces on \([0,1)\times \mathbb {R}\) ). We conclude the paper with an analogue of Gröchenig-Lyubarskii’s sufficient condition for Gabor super-frames with Hermite functions, which is equivalent to a sufficient sampling condition on the full Fock space of polyanalytic functions on \([0,1)\times \mathbb {R}\) . The proofs of the results about Gabor frames, involving some of Gröchenig’s most significant results of the past 25 years, are a clear indication of his influence on the field during this period.
