Balian-Low type theorems in finite dimensions

TL;DR

This paper develops finite-dimensional versions of the classical Balian-Low theorem and a quantitative variant, demonstrating their implications for the continuous case on the real line.

Contribution

It introduces finite-dimensional analogs of the Balian-Low theorems and proves their equivalence to the classical results on the real line.

Findings

Finite-dimensional Balian-Low theorems formulated and proved.

Quantitative Balian-Low theorem extended to finite dimensions.

Results imply classical theorems on the real line.

Abstract

We formulate and prove finite dimensional analogs for the classical Balian-Low theorem, and for a quantitative Balian-Low type theorem that, in the case of the real line, we obtained in a previous work. Moreover, we show that these results imply their counter-parts on the real line.