The Structure of Shift-Invariant Subspaces of L2(Rn)

Bownik, Marcin

doi:10.1006/jfan.2000.3635

Cited by 234 publications

(325 citation statements)

References 8 publications

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“…As the work of Ron and Shen [39] and Bownik [5] show, certain Gramian and so-called dual Gramian matrices as well as a fiberization technique play an important role in the study of TI systems. The fiberization technique is closely related to Zak transform methods in Gabor analysis, as we will see in Section 4.1.…”

Section: Fiberizationmentioning

confidence: 99%

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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: Fiberizationmentioning

confidence: 99%

“…For translation invariant systems we consider fiberization characterization of frames for translation invariant subspaces (Theorem 3.1), generalizing results from [5,7,8,39]. Using these fiberization techniques we will develop Zak transform methods for Gabor analysis in L 2 (G).…”

Section: Introductionmentioning

confidence: 99%

Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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“…The theory of shift-invariant spaces has been extensively described in the literature, e.g., [2,3,4,11], yet it will be convenient to establish a minimal amount of machinery in order to naturally develop the results of subsequent sections.…”

Section: Shift-invariant Spacesmentioning

confidence: 99%

Stable Filtering Schemes with Rational Dilations

Johnson

2007

J Fourier Anal Appl

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Abstract. The relationship between multiresolution analysis and filtering schemes is a well-known facet of wavelet theory. However, in the case of rational dilation factors, the wavelet literature is somewhat lacking in its treatment of this relationship. This work seeks to establish a means for the construction of stable filtering schemes with rational dilations through the theory of shift-invariant spaces. In particular, principal shift-invariant spaces will be shown to offer frame wavelet decompositions for rational dilations even when the associated scaling function is not refinable. Moreover, it will be shown that such decompositions give rise to stable filtering schemes with finitely supported filters, reminiscent of those studied by Kovačević and Vetterli.

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“…The closed span of G in some L p is a shift-invariant space. For shift-invariant systems, see [2,4,18,30]. On a theoretical level, the main objectives are to understand the spanning and stability properties of G. These are encoded in the spectrum of the frame operator associated to G. More generally, given the shift-invariant systems G and H = {T ak h j : k ∈ Z n , j ∈ I }, we define the frame type operator S = S G,H by…”

Section: Shift-invariant Systemsmentioning

confidence: 99%

“…The corresponding Gabor frame type operator 4) commutes with the translations T ak , for k ∈ Z n , and it follows from [14,Corollary 3.3.3 (iii)] that S is continuous from S(R n ) into S (R n ).…”

Section: Gabor Systemsmentioning

confidence: 99%