Density, overcompleteness, and localization of frames

By Radu V. Balan, Peter G. Casazza, Christopher Heil, Zeph Landau. 2006

159 citations recorded by the source corpus.

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 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. A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of F , the relative measure of 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.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Signal Processing

View Paper: https://www.semanticscholar.org/paper/951cfd82e66d3e45a11a9ce41b9d9fa20de8db0c

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In Applied Mathematics

In Computer Vision and Pattern Recognition

In Signal Processing

Citations in the corpus

Banach algebras of pseudodifferential operators and their almost diagonalization (2007, background): “ch more general systems than Gabor frames. These are so-called time-frequency molecules. They were introduced in [25], a diﬀerent version of time-frequency molecules was investigated independently in [2]. Deﬁnition 7.1. Let G(g,Λ) be a tight Gabor frame with Vgg∈ W(A). We say that {eµ ∈ Rd : µ∈ Λ} f”
Gabor Frames in Finite Dimensions (2013, background): “X and references therein), and so-called localized frames were first considered in the realm of Gabor frames [3, 4, 19, 47].”
Weight Functions in Time-Frequency Analysis (2006, methodology): “REMARK: This theorem rests very much on theorems about inverse-closedness, and can also be formulated for `(1)-conditions (as is shown in [3]) and for more general subconvolutive weights, namely precisely those satisfying the GRS-condition.”
Wiener’s Lemma for Infinite Matrices II (2007, methodology): “The classical Wiener’s lemma and its various generalizations (see, for instance, [ 3 , 4, 5, 16, 23, 24, 25, 26, 33]) are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory.”
Wiener's lemma for infinite matrices with polynomial off-diagonal decay (2005, background): “, des termes llows: rtant and 4], time- In this Note, we give a simple elementary proof to Wiener’s lemma for infinite matrices with polynomial off-diagonal”
Weight Functions in Time-Frequency Analysis (2006, background): “REMARK: This theorem rests very much on theorems about inverse-closedness, and can also be formulated for l(1)-conditions (as is shown in [3]) and for more general subconvolutive weights, namely precisely those satisfying the GRS-condition.”