Time-frequency concentration of generating systems
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- by Philippe Jaming and Alexander M. Powell PDF
- Proc. Amer. Math. Soc. 139 (2011), 3279-3290 Request permission
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 $|t|^p$ 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.References
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Additional Information
- 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
- Email: Philippe.Jaming@gmail.com
- Alexander M. Powell
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 712100
- Email: alexander.m.powell@vanderbilt.edu
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 3279-3290
- MSC (2010): Primary 42A38, 42A65
- DOI: https://doi.org/10.1090/S0002-9939-2011-10768-8
- MathSciNet review: 2811283