Balian–Low phenomenon for subspace Gabor frames

Gabardo, Jean-Pierre; Han, Deguang

doi:10.1063/1.1768621

Cited by 55 publications

(41 citation statements)

References 17 publications

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“…The main purpose of this article is to obtain characterizations for the three types of SG-duals defined above and for the uniqueness of these three types of SG-duals in term of the fibers above the space [0, 1) n . The corresponding problem for Weyl-Heisenberg (W-H) systems, where the dual systems sought are generated by both translations and modulations operators, was solved in [9] and [10]. We note, in particular that, in the so-called rational case, the problem of characterizing the uniqueness of the corresponding W-H duals of type I and type II was solved in [9] in terms of Zibulski-Zeevi matrices which are built using the Zak transform of the window function.…”

Section: Definitionmentioning

confidence: 99%

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

Hemmat

Gabardo

2007

J Fourier Anal Appl

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Given an invertible n × n matrix B and a finite or countable subset of L 2 (R n ), we consider the collection X = {φ(· − Bk) : φ ∈ , k ∈ Z n } generating the closed subspace M of L 2 (R n ). If that collection forms a frame for M, one can introduce two different types of shift-generated (SG) dual frames for X, called type I and type II SG-duals, respectively. The main distinction between them is that a SG-dual of type I is required to be contained in the space M generated by the original frame while, for a type II SG-dual, one imposes that the range of the frame transform associated with the dual be contained in the range of the frame transform associated with the original frame. We characterize the uniqueness of both types of duals using the Gramian and dual Gramian operators which were introduced in an article by Ron and Shen and are known to play an important role in the theory of shift-invariant spaces.

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Section: Definitionmentioning

confidence: 99%

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

Hemmat

Gabardo

2007

J Fourier Anal Appl

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“…BalianLow Theorems with respect to other uncertainty principles are the subject of [20,22,21]. Gabardo and Han consider the BLT for frame and Riesz sequences and show its relation to the existence of a unique dual frame in [79]. Results for the BLT with respect to general lattices in higher dimensions appear in [67,93,19].…”

Section: The Balian-low Theoremmentioning

confidence: 99%

History and Evolution of the Density Theorem for Gabor Frames

Heil

2007

J Fourier Anal Appl

176

137

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ABSTRACT. The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler-Raz biorthogonality relations, the Duality Principle, the Balian-Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.

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“…For example, it holds in higher dimensions for rather general time-frequency lattices, and also holds if one replaces "orthonormal basis" by "Riesz basis". For recent work related to the Balian-Low Theorem see [1], [5], [6], [7], [9], [10], [16], [19], [11]. The issue of sharpness or optimality in the Balian-Low Theorem was investigated in [6].…”

Section: Introductionmentioning

confidence: 99%

An Optimal Example for the Balian--Low Uncertainty Principle

Benedetto¹,

Czaja

Powell

2006

SIAM J. Math. Anal.

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Abstract. We analyze the time-frequency concentration of the Gabor orthonormal basis G(f, 1, 1) constructed by Høholdt, Jensen, and Justesen. We prove that their window function f has near optimal time and frequency localization with respect to a non-symmetric version of the Balian-Low Theorem. In particular, we show that if (p, q) = (3/2, 3), then R |t| p− |f (t)| 2 dt < ∞ and R |γ| q− | b f (γ)| 2 dγ < ∞, for 0 < ≤ 3/2, but that both integrals are infinite if = 0.

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Balian–Low phenomenon for subspace Gabor frames

Cited by 55 publications

References 17 publications

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

History and Evolution of the Density Theorem for Gabor Frames

An Optimal Example for the Balian--Low Uncertainty Principle

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