Continuous Frames in Hilbert Space

Ali, S. Twareque; Antoine, Jean-Pierre; Gazeau, Jean‐Pierre

doi:10.1006/aphy.1993.1016

Cited by 259 publications

(292 citation statements)

References 6 publications

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“…We mention that in the literature frames and discrete frames are usually called continuous frames and frames, respectively. The concept of continuous frames was introduced by Kaiser [33] and Ali, Antoine, and Gazeau [1]. For an introduction to frame theory, we refer the reader to [11].…”

Section: Frame Theorymentioning

confidence: 99%

Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler-Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames.

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Section: Frame Theorymentioning

confidence: 99%

Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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“…Later, motivated by the theory of coherent states, this concept was generalized by Antoine et al to families indexed by some locally compact space endowed with a Radon measure. Their approach leads to the notion of continuous frames [1,2,31,43]. Prominent examples are connected to the continuous wavelet transform [1,38] and the short time Fourier transform [34].…”

Section: Introductionmentioning

confidence: 99%

Continuous Frames, Function Spaces, and the Discretization Problem

Fornasier

Rauhut

2005

J Fourier Anal Appl

137

221

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A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Gröchenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.

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“…For more information on (continuous) frames, we refer to [1,2,8,18,22,34]. To a frame {f k } k∈M for H, we associate the frame transform given by…”

Section: Frame Theorymentioning

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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Continuous Frames in Hilbert Space

Cited by 259 publications

References 6 publications

Co-compact Gabor Systems on Locally Compact Abelian Groups

Co-compact Gabor Systems on Locally Compact Abelian Groups

Continuous Frames, Function Spaces, and the Discretization Problem

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

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