Frames in Banach spaces

Терехин, П. А.

doi:10.1007/s10688-010-0024-z

Cited by 17 publications

(12 citation statements)

References 19 publications

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“…The converse holds as well; each representing system in a Banach space X is also a frame in X with respect to some sequence space, which is defined, in general, in contrast to a basis, in a non-unique way (cf. [19,20]).…”

Section: Preliminariesmentioning

confidence: 99%

Representing Systems of Dilations and Translations in Symmetric Function Spaces

Асташкин

Терехин

2020

J Fourier Anal Appl

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Let X be an arbitrary separable symmetric space on [0, 1]. By using a combination of the frame approach and the notion of the multiplicator space M (X) of X with respect to the tensor product, we investigate the problem when the sequence of dyadic dilations and translations of a function f ∈ X is a representing system in the space X. The main result reads that this holds whenever 1 0 f (t) dt = 0 and f ∈ M (X). Moreover, the condition f ∈ M (X) turns out to be sharp in a certain sense. In particular, we prove that a decreasing nonnegative function f , f = 0, from a Lorentz space Λ ϕ generates an absolutely representing system of dyadic dilations and translations in Λ ϕ if and only if f ∈ M (Λ ϕ ).

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Section: Preliminariesmentioning

confidence: 99%

Representing Systems of Dilations and Translations in Symmetric Function Spaces

Асташкин

Терехин

2020

J Fourier Anal Appl

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“…Definition 2.8. [34] Let X be a Banach space and X d be a BK-space with the sequence of canonical unit vectors {e n } as basis. Let Y d be a sequence space mentioned in Lemma 2.7.…”

Section: )mentioning

confidence: 99%

Approximative K-atomic decompositions and frames in Banach spaces

Jahan

2018

AJMS

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K-frames and atomic systems for an operator K in Hilbert spaces were introduced by Gavruta [17] and further studied by Xio, Zhu and Gavruta [35]. In this paper, we have introduced the notion of an approximative atomic system for an operator K in Banach spaces and obtained interesting results. A complete characterization of family of approximative local atoms of subspace of Banach space has been obtained. Also, a necessary and sufficient condition for the existence of an approximative atomic system for an operator K is given. Finally, explicit methods are given for the construction of an approximative atomic systems for an operator K from a given Bessel sequence and approximative X d -Bessel sequence.∞ k=1 f, f k f k , f ∈ H. The frame operator S is bounded, linear and invertible on H. Thus, a frame for H allows each vector in H to be written as a linear combination of the elements in the frame, but the linear independence between the elements is not required; i.e for each vector f ∈ H we have,

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“…Let us prove that for a sequence (y n ) the inequality (1) holds. By completely repeating the proof of Theorem 4, we can obtain formula (4). Consider the analysis operator Rx = (x,…”

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confidence: 92%

Cross-frames in Banach spaces

Kreys

2011

Preprint

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This paper generalizes the results obtained in [1] for alternate dual frames in Hilbert spaces on the situation of a Banach space. Additionally, some properties of synthesis operator associated with the alternate dual frame are investigated.Definition 1. A frame for a Hibert space H is a family of vectors {ϕ n } ∞ n=1 ⊂ H \{0}, for which there are positive constants A and B, satisfying for all f ∈ H:

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Frames in Banach spaces

Cited by 17 publications

References 19 publications

Representing Systems of Dilations and Translations in Symmetric Function Spaces

Representing Systems of Dilations and Translations in Symmetric Function Spaces

Approximative K-atomic decompositions and frames in Banach spaces

Cross-frames in Banach spaces

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