Discrete and Continuous Balian-Low Theorems in Several Dimensions.

Recently, Shahaf Nitzan and Jan-Fredrik Olsen showed that Balian-Low type theorems exist for discrete Gabor systems defined on $\mathbb{Z}_d$. Here, we extend these results to higher dimensional analogs of these systems on $\mathbb{Z}_d^{l}$, and show a variety of applications of both the discrete and continuous verisons of the so-called Quantitative Balian-Low Theorem, also of Nitzan and Olsen. In particular, we prove nonsymmetric versions of the finite and continuous Balian-Low Theorems holding for $\ell_2(\mathbb{Z}_d^l)$ and $L^2(\mathbb{R}^l)$, respectively.