Banach Spaces of Distributions Defined by Decomposition Methods, I

Feichtinger, Hans G.; Gröbner, Peter

doi:10.1002/mana.19851230110

Cited by 158 publications

(226 citation statements)

References 17 publications

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“…Proof: By Proposition 3.1 in [21] we have to prove that any permutation π : I → I satisfying π(i) ⊂ i * for all i ∈ I induces a bounded operator on Y ♮ , i.e., (λ π(i) ) i∈I |Y ♮ ≤ C ′ (λ i ) i∈I |Y ♮ .…”

Section: Preparationsmentioning

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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Section: Preparationsmentioning

confidence: 99%

Continuous Frames, Function Spaces, and the Discretization Problem

Fornasier

Rauhut

2005

J Fourier Anal Appl

137

221

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“…Feichtinger has developed a general notion of amalgam spaces, e.g., [21,28,23,25,26]. For an introduction (with extensive references to the original literature) to the particular Wiener amalgams appearing in the following definition, we refer to [43].…”

Section: Remarkmentioning

confidence: 99%

Density, Overcompleteness, and Localization of Frames. II. Gabor Systems

Bălan

Casazza

Heil

et al. 2006

J Fourier Anal Appl

162

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ABSTRACT. This work develops a quantitative framework for describing the overcompleteness of a large class of frames. A previous article introduced notions of localization and approximation between two frames F = {f i } i∈I and E = {e j } 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. This article shows that those abstract results yield an array of new implications for irregular Gabor frames. Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and 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.The notions of localization and related approximation properties are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. In this article, a comprehensive examination of the interrelations among these localization and approximation concepts is made, with most implications shown to be sharp.

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“…In the above definition, instead of the family of functions (φ a ) one can use just about any bounded uniform partition of unity [21,22]. One would obtain the same space as a result [14].…”

Section: 3mentioning

confidence: 99%

Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces

Баскаков

Krishtal

2015

Mediterr. J. Math.

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Abstract. We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator L = −d/dt+A in homogeneous function spaces. We provide sufficient conditions for invertibility of such operators in terms of the spectral properties of the operator A and the semigroup generated by A. We introduce a homogeneous space of functions with absolutely summable spectrum and prove a generalization of the Gearhart-Prüss Theorem for such spaces. We use the results to prove existence and uniqueness of solutions of a certain class of non-linear equations.

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Banach Spaces of Distributions Defined by Decomposition Methods, I

Cited by 158 publications

References 17 publications

Continuous Frames, Function Spaces, and the Discretization Problem

Continuous Frames, Function Spaces, and the Discretization Problem

Density, Overcompleteness, and Localization of Frames. II. Gabor Systems

Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces

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