Balian-Low Theorems in Several Variables

TL;DR

This paper extends Balian-Low theorems to multivariable discrete and continuous Gabor systems, providing new quantitative and nonsymmetric versions, and establishing their implications in finite settings.

Contribution

It generalizes Balian-Low theorems to multiple variables and demonstrates their applications in both discrete and continuous frameworks.

Findings

Balian-Low theorems are extended to multivariable settings.

Quantitative BLTs are applied to nonsymmetric cases.

Finite BLT results are derived from quantitative versions.

Abstract

Recently, Nitzan and Olsen showed that Balian-Low theorems (BLTs) hold for discrete Gabor systems defined on $\mathbb{Z}_d$. Here we extend these results to a multivariable setting. Additionally, we show a variety of applications of the Quantitative BLT, proving in particular nonsymmetric BLTs in both the discrete and continuous setting for functions with more than one argument. Finally, in direct analogy of the continuous setting, we show the Quantitative Finite BLT implies the Finite BLT.