A critical-exponent Balian-Low theorem

Abstract

Using a variant of the Sobolev Embedding Theorem, we prove an uncertainty principle related to Gabor systems that generalizes the Balian-Low Theorem. Namely, if $f\in H^{p/2}(\mathbb{R})$ and $\hat{f}\in H^{p^{\prime }/2}(\mathbb{R})$ with $1<p<\infty$, $\frac{1}{p}+\frac{1}{p^{\prime }}=1$, then the Gabor system $\mathcal{G}(f,1,1)$ is not a frame for $L^2(\mathbb{R})$. In the $p=1$ case, we obtain a generalization of a result of Benedetto, Czaja, Powell, and Sterbenz.

14 pages, 1 figure, minor typos corrected, Remark (2) modified