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Transactions of the American Mathematical Society

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A $(p,q)$ version of Bourgain's theorem

Authors: John J. Benedetto and Alexander M. Powell
Journal: Trans. Amer. Math. Soc. 358 (2006), 2489-2505
MSC (2000): Primary 42C99
Published electronically: May 26, 2005
MathSciNet review: 2204041
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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 $n$. 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.

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Additional Information

John J. Benedetto
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Alexander M. Powell
Affiliation: Program in Applied and Computational Mathematics, Princeton University, Washington Road, Fine Hall, Princeton, New Jersey 08540

Keywords: Fourier analysis, the uncertainty principle, Gabor analysis
Received by editor(s): December 3, 2003
Received by editor(s) in revised form: May 5, 2004
Published electronically: May 26, 2005
Additional Notes: The first author wishes to acknowledge support from NSF DMS Grant 0139759. Both authors were supported in part by ONR Grant N000140210398
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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