Density, Overcompleteness, and Localization of Frames. II. Gabor Systems |
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Authors: | Radu Balan Peter G Casazza Christopher Heil Zeph Landau |
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Institution: | (1) Siemens Corporate Research, 755 College Road East, Princeton, NJ 08540, USA;(2) Department of Mathematics, University of Missouri, Columbia, MO 65211, USA;(3) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA;(4) Department of Mathematics R8133, The City College of New York, Convent Ave at 138th Street, New York, NY 10031, USA |
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Abstract: | This work develops a quantitative framework for describing the overcompleteness of a large class of frames. A previous article
introduced notions of localization and approximation between two frames F = {fi}i∈I and E = {ej}j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via
a map a : I → G. This article shows that those abstract results yield an array of new implications for irregular Gabor frames.
Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and 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. The notions of localization and related approximation properties are a spectrum
of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. In this
article, a comprehensive examination of the interrelations among these localization and approximation concepts is made, with
most implications shown to be sharp. |
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