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A version of Bourgain's theorem

Authors:	John J Benedetto Alexander M Powell

Institution:	Department of Mathematics, University of Maryland, College Park, Maryland 20742 ; Program in Applied and Computational Mathematics, Princeton University, Washington Road, Fine Hall, Princeton, New Jersey 08540

Abstract:	Let $1<p,q<\infty$ satisfy $\frac{1}{p} + \frac{1}{q} =1.$ We construct an orthonormal basis $\{ b_n \}$ for $L^2 (\mathbb{R})$ such that $\Delta_p ( b_n )$ and $\Delta_q (\widehat{b_n})$ are both uniformly bounded in . Here $\Delta_{\lambda} (f) \equiv {\rm inf}_{a \in \mathbb{R}} \left( \int \vert x - a\vert^{\lambda} \vert f(x)\vert^2 dx \right)^{\frac{1}{2}}$ . This generalizes a theorem of Bourgain and is closely related to recent results on the Balian-Low theorem.

Keywords:	Fourier analysis the uncertainty principle Gabor analysis

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