A $(p,q)$ version of Bourgainâs theorem
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- by John J. Benedetto and Alexander M. Powell PDF
- Trans. Amer. Math. Soc. 358 (2006), 2489-2505 Request permission
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 $\Delta _{\lambda } (f) \equiv \textrm {inf}_{a \in \mathbb {R}} \left ( \int |x - a|^{\lambda } |f(x)|^2 dx \right )^{\frac {1}{2}}$. This generalizes a theorem of Bourgain and is closely related to recent results on the Balian-Low theorem.References
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Additional Information
- John J. Benedetto
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- Email: jjb@math.umd.edu
- Alexander M. Powell
- Affiliation: Program in Applied and Computational Mathematics, Princeton University, Washington Road, Fine Hall, Princeton, New Jersey 08540
- MR Author ID: 712100
- Email: apowell@math.princeton.edu
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 2489-2505
- MSC (2000): Primary 42C99
- DOI: https://doi.org/10.1090/S0002-9947-05-03737-2
- MathSciNet review: 2204041