Mathematical Research Letters

Volume 13 (2006)

Number 3

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

Pages: 467 – 474



John J. Benedetto (University of Maryland, College Park)

Wojciech Czaja (University of Vienna)

Alexander M. Powell (Vanderbilt University)

Jacob Sterbenz (University of California at San Diego)


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 $\Delta_{\infty} (\widehat{g}) = {\rm sup}_{N >0} \int |\g|^N |\widehat{g} (\g)|^2 d\g.$

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