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Time-frequency concentration of generating systems

Authors: Philippe Jaming and Alexander M. Powell
Journal: Proc. Amer. Math. Soc. 139 (2011), 3279-3290
MSC (2010): Primary 42A38, 42A65
Published electronically: February 9, 2011
MathSciNet review: 2811283
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Abstract: Uncertainty principles for generating systems $ \{e_n\}_{n=1}^{\infty} \subset L^2(\mathbb{R})$ are proven and quantify the interplay between $ \ell^r(\mathbb{N})$ coefficient stability properties and time-frequency localization with respect to $ \vert t\vert^p$ power weight dispersions. As a sample result, it is proven that if the unit-norm system $ \{e_n\}_{n=1}^{\infty}$ is a Schauder basis or frame for $ L^2(\mathbb{R})$, then the two dispersion sequences $ \Delta(e_n)$, $ \Delta(\widehat{e_n})$ and the one mean sequence $ \mu(e_n)$ cannot all be bounded. On the other hand, it is constructively proven that there exists a unit-norm exact system $ \{f_n\}_{n=1}^{\infty}$ in $ L^2(\mathbb{R})$ for which all four of the sequences $ \Delta(f_n)$, $ \Delta(\widehat{f_n})$, $ \mu(f_n)$, $ \mu(\widehat{f_n})$ are bounded.

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Philippe Jaming
Affiliation: Faculté des Sciences, MAPMO, Université d’Orléans, BP 6759, F 45067 Orléans Cedex 2, France
Address at time of publication: Institut de Mathématiques de Bordeaux UMR 5251, Université Bordeaux 1, cours de la Libération, F 33405 Talence cedex, France

Alexander M. Powell
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

Keywords: Compactness, exact system, frame, Schauder basis, time-frequency concentration, uncertainty principle.
Received by editor(s): February 22, 2010
Received by editor(s) in revised form: August 23, 2010
Published electronically: February 9, 2011
Additional Notes: The second author was supported in part by NSF Grant DMS-0811086. Portions of this work were completed during visits to the Université d’Orléans (Orléans, France), the Academia Sinica Institute of Mathematics (Taipei, Taiwan), and the City University of Hong Kong (Hong Kong, China). The second author is grateful to these institutions for their hospitality and support. The first and second authors were both partially supported by the ANR project AHPI Analyse Harmonique et Problèmes Inverses
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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