Mathematical Research Letters

Volume 13, Issue 3, May 2006 pp. 467-474.

An Endpoint $(1,\infty)$ Balian-Low Theorem

Authors: John J. Benedetto (1), Wojciech Czaja (2) Alexander M. Powell (3), and Jacob Sterbenz (4)
Author institution: University of Maryland, College Park (1), University of Vienna (2) Vanderbilt University (3), and University of California, San Diego (4)

Summary: It is shown that a $(1, \infty)$ version of the Balian-Low Theorem holds. If $g \in L^2(\linR),$ $\Delta_1 ({g}) < \infty$ and $\Delta_{\infty} (\widehat{g}) < \infty,$ then the Gabor system $\mathcal{G} (g,1,1)$ is not a Riesz basis for $L^2(\linR)$. Here, $\Delta_1 ({g}) = \int |t| |g(t)|^2 dt$ and $\Delta_{\infty} (\widehat{g}) = {\rm sup}_{N>0} \int |\g|^N |\widehat{g} (\g)|^2 d\g.$

Contents Full-Text PDF