Density, Overcompleteness, and Localization of Frames. I. Theory |
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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: | Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes
them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition
that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative
framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two
frames
and
(
a discrete
abelian group), relating the decay of the expansion of the elements of
in terms of the elements of
via a map
. A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of
, the relative measure of
— 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
in
. 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. In a subsequent article, these results are applied
to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The
notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify
the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of
the interrelations among these localization and approximation concepts is presented. |
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Keywords: | |
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