Weaving K-Frames in Hilbert Spaces

Deepshikha,; Vashisht, Lalit Kumar

doi:10.1007/s00025-018-0843-4

Cited by 33 publications

(16 citation statements)

References 28 publications

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“…where the second inequality is deduced by (19). Hence {Λ j : j ∈ J} and {Γ j : j ∈ J} are K-woven in R(K).…”

Section: Theoremmentioning

confidence: 92%

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Weaving of K-g-frames in Hilbert spaces

Xiao¹,

Zhou²,

Zhu³

2019

ScienceAsia

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In this study, we mainly discuss the weaving of K-g-frames in Hilbert spaces. Note that the concept of weaving was recently proposed by Bemrose et al to solve a question in distributed signal processing. We give a sufficient condition such that the left sequence can still be K-woven in R(K) by deleting some elements from a K-woven pair of K-g-frames. We then give three different types of perturbation conditions such that, under them, the K-g-frames {Λ j : j ∈ J} and {Γ j : j ∈ J} or the types {Λ j T * 1 : j ∈ J} and {Γ j T * 2 : j ∈ J} can be K-woven in R(K), where T 1 , T 2 are surjective operators on .

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“…where the second inequality is deduced by (19). Hence {Λ j : j ∈ J} and {Γ j : j ∈ J} are K-woven in R(K).…”

Section: Theoremmentioning

confidence: 92%

“…15-18 for more information on the weaving of g-frames or fusion frames and Ref. 19 for information on the weaving of K-frames.…”

Section: Introductionmentioning

confidence: 99%

Weaving of K-g-frames in Hilbert spaces

Xiao¹,

Zhou²,

Zhu³

2019

ScienceAsia

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“…After that, Vashisht, Deepshikha, Arefijamaal and etc. have done more studies over the past few years [1,9,26,27].…”

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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“…The theory may be applied to deal with wireless sensor networks that require distributed processing under different frames, which could also be used in the pre-processing of signals by means of Gabor frames. Recently, weaving frames have attracted many scholars' attention, please refer to [24][25][26][27][28][29][30] for more information.…”

Section: Introductionmentioning

confidence: 99%

More on Inequalities for Weaving Frames in Hilbert Spaces

Xiang

2019

Mathematics

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In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real numbers. It is indicated that our results are more general and cover the corresponding results recently obtained by Li and Leng. We also give a triangle inequality for weaving frames in Hilbert spaces, which is structurally different from previous ones.

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Weaving K-Frames in Hilbert Spaces

Cited by 33 publications

References 28 publications

Weaving of K-g-frames in Hilbert spaces

Weaving of K-g-frames in Hilbert spaces

Frame-related operators for woven frames

More on Inequalities for Weaving Frames in Hilbert Spaces

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