G-frames with bounded linear operators

Xiao, Xiangchun; Zhu, Yucan; Zhibiao, Shu; Ding, Ming-ling

doi:10.1216/rmj-2015-45-2-675

Cited by 23 publications

(10 citation statements)

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“…Suppose that {Λ j : j ∈ J} and {Γ j : j ∈ J} are g-frames for with respect to { j : j ∈ J} with the frame bounds A 1 , B 1 and A 2 , B 2 , respectively. If there exist α, β, γ ∈ [0, ∞) satisfying A 1 > αB 1 + β B 2 + γ such that, for any {g j } j∈J ∈ l 2…”

Section: Corollarymentioning

confidence: 99%

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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Section: Corollarymentioning

confidence: 99%

Weaving of K-g-frames in Hilbert spaces

Xiao¹,

Zhou²,

Zhu³

2019

ScienceAsia

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“…In recent years, several generalizations of frames in Hilbert and Banach spaces have been proposed and studied like fusion frames in Hilbert spaces [6], g-frames by Sun [7,8], fusion Banach frames in Banach spaces by , K-frames by Găvruta [12,13], K-g-frames by Xiao et al [14], and scalable K-frames by Ramesan and Ravindran [15]. The concept of K-frames was put forward to study the atomic systems with respect to a bounded linear operator K. These frames allowed to reconstruct elements from the range of bounded linear operator K and were more flexible than the classical frames.…”

Section: Introductionmentioning

confidence: 99%

“…Further, it has been seen that K-g-frames are the generalization of g-frames and have better practical applications. For more information on K-g-frames, one may refer to [14,[16][17][18]. Tight g-frames are similar to tight frames and are helpful in reconstructing signals.…”

Section: Introductionmentioning

confidence: 99%

Scalability of Generalized Frames for Operators

Kumar,

Malhotra,

Khanna

2024

Journal of Function Spaces

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In this paper, the Parseval K-g-frames are constructed from a given K-g-frame by scaling the elements of the K-g-frame with the help of diagonal operators, and these frames are named scalable K-g-frames. Also, we prove some properties of scalable K-g-frames and construct new scalable K-g-frames from a given K-g-frame. The necessary and sufficient conditions for a K-g-frame to be scalable are given. Further, equivalent conditions for the scalability of K-g-frames and the K-frames induced by K-g-frames are obtained. Finally, it is shown that the direct sum of two scalable K-g-frames is again a scalable K-g-frame for some suitable bounded linear operator K.

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“…where K † is the pseudo inverse of K, see [7]. For further information on K-frames refer to [15,16].…”

Section: Introductionmentioning

confidence: 99%

Some constructions of $K$-frames and their duals

Neyshaburi¹,

Arefijamaal²

2017

Rocky Mountain J. Math.

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K-frames, as a new generalization of frames, have important applications, especially in sampling theory, to help us to reconstruct elements from a range of a bounded linear operator K in a separable Hilbert space. In this paper, we focus on the reconstruction formulae to characterize all K-duals of a given K-frame. Also, we give several approaches for constructing K-frames.2010 AMS Mathematics subject classification. Primary 42C15, Secondary 41A58.

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G-frames with bounded linear operators

Cited by 23 publications

References 0 publications

Weaving of K-g-frames in Hilbert spaces

Weaving of K-g-frames in Hilbert spaces

Scalability of Generalized Frames for Operators

Some constructions of $K$-frames and their duals

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