Frame properties of wave packet systems in $L^2({\mathbb R}^d)$

Christensen, Ole; Rahimi, Asghar

doi:10.1007/s10444-007-9038-3

Cited by 49 publications

(28 citation statements)

References 14 publications

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“…Proposition 3.7 is a generalization of the results in, e.g., [10] and [9], which state the corresponding result for GSI systems in the euclidean space and locally compact abelian groups, respectively. The result is as follows.…”

Section: On Sufficient Conditions and The Local Integrability Conditionsmentioning

confidence: 99%

Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

Jakobsen

Lemvig

2016

Trans. Amer. Math. Soc.

104

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In this paper we connect the well established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let Γ j , j ∈ J, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian group G and study systems of the form ∪ j∈J {g j,p (· − γ)} γ∈Γj,p∈Pj with generators g j,p in L 2 (G) and with each P j being a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continuous variants, are GTI systems. Under a technical α local integrability condition (α-LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas for L 2 (G). This generalizes results on generalized shift invariant systems, where each P j is assumed to be countable and each Γ j is a uniform lattice in G, to the case of uncountably many generators and (not necessarily discrete) closed, co-compact subgroups. Furthermore, even in the case of uniform lattices Γ j , our characterizations improve known results since the class of GTI systems satisfying the α-LIC is strictly larger than the class of GTI systems satisfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames for L 2 (G). In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor transform in L 2 (R n ) are special cases of the same general characterizing equations.

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Section: On Sufficient Conditions and The Local Integrability Conditionsmentioning

confidence: 99%

Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

Jakobsen

Lemvig

2016

Trans. Amer. Math. Soc.

104

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“…Proposition IV.1 is a generalization of the results in, e.g., [20] and [21], which state the corresponding result for generalized shift invariant systems in the euclidean space and locally compact abelian groups. The result is as follows: (ii) Furthermore, if also…”

Section: Conditionsmentioning

confidence: 99%

A characterization of tight and dual generalized translation invariant frames

Jakobsen

Lemvig

2015

2015 International Conference on Sampling Theory and Applications (SampTA)

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Abstract-We present results concerning generalized translation invariant (GTI) systems on a second countable locally compact abelian group G. These are systems with a family of generators {gj,p}j∈J,p∈P j ⊂ L 2 (G), where J is a countable index set, and Pj, j ∈ J are certain measure spaces. Furthermore, for each j we let Γj be a closed subgroup of G such that G/Γj is compact. A GTI system is then the collection of functions ∪j∈J {gj,p(· − γ)}γ∈Γ j ,p∈P j . Many well known systems, such as wavelet, shearlet and Gabor systems, both the discrete and continuous types, are GTI systems. We characterize when such systems form tight frames, and when two GTI Bessel systems form dual frames for L 2 (G). In particular, this offers a unified approach to the theory of discrete and continuous frames and, e.g., yields well known results for discrete and continuous Gabor and wavelet systems.

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“…In particular, they showed necessary conditions for the wave packet system to be a Bessel system. In 2008, Christensen and Rahimi [4] considered wave packet systems as special cases of generalized shift-invariant systems and presented a sufficient condition for a wave packet system to form a frame. They also presented certain natural conditions on the parameters in a wave packet system which exclude the frame property.…”

Section: Introductionmentioning

confidence: 99%