Wiener Amalgam Spaces for the Fundmental Identity of Gabor Analysis

TL;DR

This paper explores the Fundamental Identity of Gabor Analysis through the lens of Wiener amalgam spaces, extending its validity and connecting it with operator algebras and time-frequency analysis.

Contribution

It introduces a new description of the validity range of the Fundamental Identity of Gabor Analysis using Wiener amalgam spaces and Feichtinger's algebra, inspired by Rieffel's operator algebra approach.

Findings

Extended the Fundamental Identity's validity range using Wiener amalgam spaces.

Connected Gabor analysis with operator algebras via the Fundamental Identity.

Provided a unified framework inspired by Rieffel's approach.

Abstract

In the last decade it has become clear that one of the central themes within Gabor analysis (with respect to general time-frequency lattices) is a duality theory for Gabor frames, including the Wexler-Raz biorthogonality condition, the Ron-Shen's duality principle or Janssen's representation of a Gabor frame operator. All these results are closely connected with the so-called {\it Fundamental Identity of Gabor Analysis}, which we derive from an application of Poisson's summation formula for the symplectic Fourier transform. The new aspect of this presentation is the description of range of the validity of this Fundamental Identity of Gabor Analysis using Wiener amalgam spaces and Feichtinger's algebra $S_0(\R^d)$. Our approach is inspired by Rieffel's use of the Fundamental Identity of Gabor Analysis in the study of operator algebras generated by time-frequency shifts along a lattice, which was later independently rediscovered by Tolmieri/Orr, Janssen, and Daubechies et al., and Feichtinger/Kozek at various levels of generality, in the context of Gabor analysis.