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ISSN 1079-6762


Density, overcompleteness, and localization of frames

Authors: Radu Balan, Peter G. Casazza, Christopher Heil and Zeph Landau
Journal: Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 71-86
MSC (2000): Primary 42C15; Secondary 46C99
Published electronically: July 7, 2006
MathSciNet review: 2237271
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Abstract: This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames $ \mathcal{F} = \{f_i\}_{i \in I}$ and $ \mathcal{E} = \{e_j\}_{j \in G}$ ($ G$ a discrete abelian group), relating the decay of the expansion of the elements of $ \mathcal{F}$ in terms of the elements of $ \mathcal{E}$ via a map $ a \colon I \to G$. A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of $ \mathcal{F}$, the relative measure of $ \mathcal{E}$--both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements--and the density of the set $ a(I)$ in $ G$. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. These abstract results yield an array of new implications for irregular Gabor frames. Various Nyquist density results for Gabor frames are recovered as special cases, but in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the time-frequency plane.

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Additional Information

Radu Balan
Affiliation: Siemens Corporate Research, 755 College Road East, Princeton, New Jersey 08540

Peter G. Casazza
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Christopher Heil
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Zeph Landau
Affiliation: Department of Mathematics R8133, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031

PII: S 1079-6762(06)00163-6
Keywords: Density, excess, frames, Gabor systems, modulation spaces, overcompleteness, Riesz bases, wavelets, Weyl--Heisenberg systems.
Received by editor(s): July 10, 2005
Published electronically: July 7, 2006
Additional Notes: The second author was partially supported by NSF Grants DMS-0102686 and DMS-0405376.
The third author was partially supported by NSF Grant DMS-0139261.
The fourth author was partially supported by The City University of New York PSC-CUNY Research Award Program.
Communicated by: Guido Weiss
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.