Gabor frames for quasi-periodic functions and polyanalytic spaces on the flat cylinder

Abstract

We develop an alternative approach to the study of Fourier series, based on the Short-Time-Fourier Transform (STFT) acting on $L_{\hat{I}½}^2(0,1)$, the space of measurable functions $f$ in $R$, square-integrable in $(0,1)$, and time-periodic up to a phase factor: for fixed $\hat{I}½\in \mathbb{R}$, \begin{equation*} f(t+k)=e^{2πikν}f(t)\text{, }k\in \mathbb{Z}\text{.} \end{equation*} The resulting phase space is $[0,1)\times R$, a flat model of an infinite cylinder, leading to Gabor frames with a rich structure, including a Janssen-type representation. A Gaussian window leads to a Fock space of entire functions, studied in the companion paper by the same authors [\emph{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 $[0,1)\times R$. Furthermore, an analogue of the sufficient Wexler-Raz conditions is obtained. This leads to a new criteria for Gabor frames in $L^2(R)$, to sufficient conditions for Gabor frames in $L_{\hat{I}½}^2(0,1)$ with Hermite windows (an analogue of a theorem of Gröchenig and Lyubarskii about Gabor frames with Hermite windows) and with totally positive windows. We also consider a vectorial STFT in $L_{\hat{I}½}^2(0,1)$ and the (full) Fock spaces of polyanalytic functions on $[0,1)\times R$, associated Bargmann-type transforms, and an analogue of Vasilevski's orthogonal decomposition into true polyanalytic Fock spaces (Landau level eigenspaces on $[0,1)\times R$). We conclude with an analogue of Gröchenig-Lyubarskii's sufficient condition for Gabor super-frames with Hermite functions, equivalent to a sufficient sampling condition on the full Fock space of polyanalytic functions on $[0,1)\times \mathbb{R}$.

42 pages