Weaving frames

Bemrose, Travis; Casazza, Peter G.; Gröchenig, Karlheinz; Lammers, Mark; Lynch, Richard G.

doi:10.7153/oam-10-61

Cited by 72 publications

(78 citation statements)

References 9 publications

(10 reference statements)

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“…We extend this result in the context of g-Riesz basis. The idea of the proof is similar to Theorem 5.3 in [8]. It is well known that the concept of ordinary Riesz basis and ordinary exact frames are equivalent.…”

Section: Weaving G-riesz Basesmentioning

confidence: 88%

“…It was proved in [8] that if a family of Riesz bases are weaving frames, then each weaving is also a Riesz basis. Some basic properties of weaving frames were studied in [9].…”

Section: Weaving Framesmentioning

confidence: 99%

“…al. [8] proved if the family of Riesz bases are woven frames, then each weaving is not only a frame but also a Riesz basis. We extend this result in the context of g-Riesz basis.…”

Section: Weaving G-riesz Basesmentioning

confidence: 99%

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Weaving K-Frames in Hilbert Spaces

Deepshikha

Vashisht

2018

Results Math

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Generalized frames (g-frames) are natural generalization of standard frames whereas generalized frames for operators (Θ-g-frames) are g-frames in which the lower frame inequality is controlled by a linear bounded operator Θ on a Hilbert space. In this article, we study weaving properties of g-frames and Θ-g-frames. We present some sufficient conditions for Θ-g-frames to be woven. It is shown that the family of Θ-g-frames obtained by applying bounded operators (whose adjoint operator is surjective) to a family of weaving Θ-g-frames is woven. A class of weaving generalized frames is established by generalized frame operator. A Paley-Weiner type perturbation result for weaving Θ-g-frames is presented. Finally, we show that weaving g-Riesz bases need not behave same as weaving ordinary Riesz bases.

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Section: Weaving G-riesz Basesmentioning

confidence: 88%

“…It was proved in [8] that if a family of Riesz bases are weaving frames, then each weaving is also a Riesz basis. Some basic properties of weaving frames were studied in [9].…”

Section: Weaving Framesmentioning

confidence: 99%

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Weaving K-Frames in Hilbert Spaces

Deepshikha

Vashisht

2018

Results Math

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“…Very recently, Bemrose et al [6] introduced a new concept of "weaving frames" in separable Hilbert spaces which is motivated by the following question in distributed signal processing: given are two sets {f j } j∈J and {g j } j∈J of linear measurements with stable recovery guarantees, in mathematical terminology each set is a frame labeled by a node or sensor j ∈ J. At each sensor a signal f was measured either with f j or with g j , so that the collected information is the set of numbers { f, f j } j∈σ ∪ { f, g j } j∈σ c for some subset σ ⊆ J.…”

Section: Weaving Framesmentioning

confidence: 98%

Controlled weaving frames in Hilbert spaces

Rezapour

Rahimi

Osgooei

et al. 2019

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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In this paper, we first introduce the notion of controlled weaving K-g-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving K-g-frames in separable Hilbert spaces. Also, a characterization of controlled weaving K-g-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled K-g-woven frames gives ordinary weaving K-frames and vice-versa.

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“…Woven frames. Woven frames in Hilbert spaces were introduced by Bemrose et al [2,4,6] in 2015. After that, Vashisht, Deepshikha, Arefijamaal and etc.…”

Section: 2mentioning

confidence: 99%

Frame-related operators for woven frames

Rahimi

Samadzadeh

Daraby

2019

Int. J. Wavelets Multiresolut Inf. Process.

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A new notion in frame theory has been introduced recently under the name woven-weaving frames by Bemrose et. al. In the studying of frames, some operators like analysis, synthesis, Gram and frame operator play the central role. In this paper, for the first time, we introduce and define these operators for woven-weaving frames and review some properties of them. In continuation, we investigate the effect of different types of operators on the woven frames and their bounds. Also, we provide some conditions that shows sum of woven frames are also woven frames. Finally, we apply these properties with an example. IntroductionThe theory of frames plays an important role in signal processing because of their resilience to quantization [15], resilience to additive noise, as well as their numerical stability of reconstruction and greater freedom to capture signal characteristics. Also frames have been used in sampling theory to oversampled perfect reconstruction filter banks, system modeling, neural networks and quantum measurements [12]. New applications in image processing, robust transmission over the internet and wireless [16], coding and communication [23] were given.Discrete frames in Hilbert spaces has been introduced by Duffin and Schaeffer [11] and popularized by Daubechies, Grossmann and Meyer [10]. A discrete frame is a countable family of elements in a separable Hilbert space which allows stable and not necessarily unique decompositions of arbitrary elements in an expansion of frame elements.The last two decades have seen tremendous activity in the development of frame theory and many generalizations of frames have come into existence which include bounded quasi-projectors [13], fusion frames [5], pseudoframes [19], oblique frames [8], g-frames [24], continuous frames [21], Kframes [14], fractional calculus [17, 18], Hilbert-Schmidt frames [22] and etc.In one of the direction of applications of frames in signal processing, a new concept of woven-weaving frames in a separable Hilbert space introduced by

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Weaving frames

Cited by 72 publications

References 9 publications

Weaving K-Frames in Hilbert Spaces

Weaving K-Frames in Hilbert Spaces

Controlled weaving frames in Hilbert spaces

Frame-related operators for woven frames

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