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

TL;DR

This paper develops a novel framework using the Short-Time Fourier Transform on a flat cylindrical phase space to analyze Gabor frames, leading to new criteria and decompositions for frames with various windows, including Hermite functions.

Contribution

It introduces a new approach to Gabor frame analysis on a flat cylinder, connecting it with polyanalytic Fock spaces and establishing novel frame conditions and decompositions.

Findings

Established a Janssen-type representation for Gabor frames on the flat cylinder.

Derived new criteria for Gabor frames with Hermite and totally positive windows.

Extended orthogonal decompositions to polyanalytic Fock spaces on the cylinder.

Abstract

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 ${R}$, square-integrable in $ (0,1)$, and time-periodic up to a phase factor: for fixed $\nu \in \mathbb{R}$, \begin{equation*} f(t+k)=e^{2\pi ik\nu }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 $L_{\nu }^{2}(0,1)$ with Hermite windows (an analogue of a theorem of Gr\"{o}chenig and Lyubarskii about Gabor frames with Hermite windows) and with totally positive windows. We also consider a vectorial STFT in $L_{\nu }^{2}(0,1)$ 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 Gr\"{o}chenig-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 $[0,1)\times \mathbb{R}$.