Some Properties of K-Frames in Hilbert Spaces

Xiao, Xiangchun; Zhu, Yixin; Gǎvruţa, Laura

doi:10.1007/s00025-012-0266-6

Cited by 96 publications

(67 citation statements)

References 16 publications

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“…for all f ∈ H. Therefore, discrete atomic systems can be regarded as special cases of our definition and many basic properties can be derived within this more general context; A similar claim is true for pseudoframes [31,32,42], frames for operators [27,40], system of bounded quasiprojections [24] and atomic resolution of the identity [5,6,9]. But continuous atomic systems as the more general setting have much difference with the discrete version.…”

Section: The Definition and Basic Resultsmentioning

confidence: 96%

“…Some interesting examples of these notions are given by [23, Proposition 5 and Proposition 6], [27,Theorem 5] and [9,19,24,31,32,40,42]. The reader will remark that the discrete version of the above definition, that is, the case that X = N and μ is the counting measure can be found in [9, Definition 3.24 and Definition 3.29].…”

Section: The Definition and Basic Resultsmentioning

confidence: 99%

“…For basic results on frames, we refer the reader to the references [10,12,14,21,22,29,30]. Here, it should be noted that H which satisfied in the equality (2) is important in the theory of frames and in many situations, the study of such family of operators is of considerable importance; see for example, [3,[5][6][7]9,11,19,20,23,25,27,28,31,32,[34][35][36][37][38][39][40]42,43]. Therefore, it is expected that the study of the family of bounded operators which satisfied in the equality (3) plays a crucial role in the theory of frames in Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Continuous Atomic Systems for Subspaces

Javanshiri

Fattahi

2015

Mediterr. J. Math.

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This paper gives new contributions to atomic systems theory in Hilbert spaces. More precisely, we introduce and study the concept of continuous atomic systems for subspaces, and we give some examples to show differences between this and the discrete version. Moreover, we give a new characterization of continuous (resp. discrete) atomic systems. Finally, among other things, we discuss the relationship between continuous (resp. discrete) frame of subspaces (fusion frames) and continuous (resp. discrete) atomic systems. In particular, we show that the perturbation of a Parseval continuous (resp. discrete) frame of subspaces or identity operator, under certain conditions, gives us a continuous (resp. discrete) atomic system. Mathematics Subject Classification. 42C15, 41A58, 47A58.

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Section: The Definition and Basic Resultsmentioning

confidence: 96%

Section: The Definition and Basic Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Continuous Atomic Systems for Subspaces

Javanshiri

Fattahi

2015

Mediterr. J. Math.

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“…Moreover, the roles of the dual K-frame pair cannot be interchanged in general (see Example 3.2 in Ref. 10), and a K-frame does not admit a dual frame in general (see Example 4 in this paper). For more details on K-frames, see Refs.…”

Section: Letmentioning

confidence: 96%

Frame sequences and dual frames for operators

Xiang¹,

Li²

2016

ScienceAsia

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ABSTRACT:We introduce the more general frame sequences and dual frames related to a linear bounded operator K in Hilbert spaces which we call K-frame sequences and dual K-frames, respectively. We give several equivalent characterizations for K-frame sequences. We also investigate the relationships among K-frame sequences, K-frames, and frame sequences, and give a new perturbation result for K-frames by using the associated dual K-frames. It turns out that in many ways K-frame sequences and dual K-frames behave completely differently from frame sequences and dual frames, respectively.

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“…K-frames were proposed by Gȃvruta [11] in Hilbert spaces to study atomic systems. In [6,9,10,19], some properties and conclusions of K-frames were given. In [22], the authors put forward the concept of K-g-frames, which are more general than ordinary g-frames in Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

K-G-Frames and Stability of K-G-Frames in Hilbert Spaces

Hua¹,

Huang²

2016

Journal of the Korean Mathematical Society

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Abstract. A K-g-frame is a generalization of a g-frame. It can be used to reconstruct elements from the range of a bounded linear operator K in Hilbert spaces. K-g-frames have a certain advantage compared with g-frames in practical applications. In this paper, the interchangeability of two g-Bessel sequences with respect to a K-g-frame, which is different from a g-frame, is discussed. Several construction methods of K-g-frames are also proposed. Finally, by means of the methods and techniques in frame theory, several results of the stability of K-g-frames are obtained.

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Some Properties of K-Frames in Hilbert Spaces

Cited by 96 publications

References 16 publications

Continuous Atomic Systems for Subspaces

Continuous Atomic Systems for Subspaces

Frame sequences and dual frames for operators

K-G-Frames and Stability of K-G-Frames in Hilbert Spaces

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