Bessel duality of Gabor systems: A von Neumann algebraic perspective

TL;DR

This paper reveals that Bessel duality in Gabor systems can be understood through von Neumann algebra bimodule theory, connecting time-frequency analysis with operator algebra concepts.

Contribution

It establishes a novel von Neumann algebraic framework to explain Bessel duality in Gabor systems, bridging two mathematical areas.

Findings

Bessel duality follows from bimodule vector properties

Connection between Gabor analysis and von Neumann algebras

Provides a new perspective on time-frequency analysis

Abstract

Bessel duality of regular Gabor systems states that a Gabor system over a lattice is a Bessel sequence if and only if the corresponding Gabor system over the adjoint lattice is a Bessel sequence. We show that this fundamental result of time-frequency analysis can be deduced from a theorem in the theory of bimodules over von Neumann algebras, namely that under certain conditions, their left and right bounded vectors coincide.