Gabor Frames: Characterizations and Coarse Structure

TL;DR

This survey systematically consolidates key characterizations of Gabor frames over lattices, emphasizing the duality theorem and providing streamlined proofs to unify the structure theory in a concise manner.

Contribution

It offers a unified, streamlined exposition of Gabor frame characterizations centered on the duality theorem, simplifying the existing structure theory.

Findings

Most characterizations derive from the duality theorem.

The proof sequence is streamlined and concise.

Many known characterizations are shown as corollaries.

Abstract

This survey offers a systematic and streamlined exposition of the most important characterizations of Gabor frames over a lattice. The goal is to collect the most important characterizations of Gabor frames and offer a systematic exposition of these structures. In the center of these characterizations is the duality theorem for Gabor frames. Most characterizations within the $L^2$-theory follow directly from this fundamental duality. In particular, the celebrated characterizations of Janssen and Ron-Shen are consequences of the duality theorem, and the characterization of Zeevi and Zibulski for rational lattices also becomes a corollary. The novelty is the streamlined sequence of proofs, so that most of the structure theory of Gabor frames fits into a single, short article. The only prerequisite is the thorough mastery of the Poisson summation formula and some basic facts about frames and Riesz sequences.