Gabor systems and almost periodic functions

TL;DR

This paper constructs Gabor frames for certain subspaces of almost periodic functions and establishes a generalized frame for the entire space, advancing the understanding of time-frequency analysis in non-separable Hilbert spaces.

Contribution

It introduces a new construction of Gabor frames for separable subspaces of almost periodic functions and identifies a non-countable generalized frame for the whole space.

Findings

Gabor frames are constructed for subspaces of $AP_2(\mathbb{R})$.

A non-countable generalized frame for $AP_2(\mathbb{R})$ is established.

Bessel-type estimates are shown for the $AP$ norm with countable Gabor systems.

Abstract

We give a construction of Gabor type frames for suitable separable subspaces of the non-separable Hilbert spaces $AP_2({\mathbb R})$ of almost periodic functions of one variable. Furthermore we determine a non-countable generalized frame for the whole space $AP_2({\mathbb R}).$ We show furthermore that Bessel-type estimates hold for the $AP$ norm with respect to a countable Gabor system using suitable almost periodic norms of sequencies.