Operator-isomorphism pairs and Zak transform methods for the study of Gabor systems

TL;DR

This paper explores the unitary equivalence of Gabor systems using operator pairs and Zak transform methods, focusing on Hermite functions and discussing double over-sampling.

Contribution

It introduces new proofs for existing results and applies Zak transform techniques to analyze Gabor systems with Hermite functions.

Findings

Unitary equivalence characterized by operator pairs and isometries

New proofs of known results on Gabor systems

Insights into double over-sampling in Gabor analysis

Abstract

We collect and summarize results on the unitary equivalence of Gabor systems by pairs of unitary operators and global isometries. The methods are then used to study Gabor systems with Hermite functions. We provide new proofs of some known results and an outlook on double over-sampling.