Abstract:This paper introduces the concept of atomic subspaces with respect to a bounded linear operator. Atomic subspaces generalize fusion frames and this generalization leads to the notion of K-fusion frames. Characterizations of K-fusion frames are discussed. Various properties of K-fusion frames, for example, direct sum, intersection, are studied.
“…Definition 2.6. [1] Let { W j } j ∈ J be a family of closed subspaces of H and { v j } j ∈ J be a family of positive weights and K ∈ B ( H ). Then { ( W j , v j ) : j ∈ J } is said to be an atomic subspace of H with respect to K if following conditions hold:…”
Section: Preliminariesmentioning
confidence: 99%
“…P. Casazza [2] was first to introduce the notion of fusion frames or frames of subspaces and gave various ways to obtain a resolution of the identity operator from a fuison frame. The concept of an atomic subspace with respect to a bounded linear operator were introduced by A. Bhandari and S. Mukherjee [1]. Construction of K-g-fusion frames and their dual were presented by Sadri and Rahimi [14] to generalize the theory of K-frame, fusion frame and g-frame.…”
We introduce the notion of a g-atomic subspace for a bounded linear operator and construct several useful resolutions of the identity operator on a Hilbert space using the theory of g-fusion frames. Also we shall describe the concept of frame operator for a pair of g-fusion Bessel sequences and some of their properties.
“…Definition 2.6. [1] Let { W j } j ∈ J be a family of closed subspaces of H and { v j } j ∈ J be a family of positive weights and K ∈ B ( H ). Then { ( W j , v j ) : j ∈ J } is said to be an atomic subspace of H with respect to K if following conditions hold:…”
Section: Preliminariesmentioning
confidence: 99%
“…P. Casazza [2] was first to introduce the notion of fusion frames or frames of subspaces and gave various ways to obtain a resolution of the identity operator from a fuison frame. The concept of an atomic subspace with respect to a bounded linear operator were introduced by A. Bhandari and S. Mukherjee [1]. Construction of K-g-fusion frames and their dual were presented by Sadri and Rahimi [14] to generalize the theory of K-frame, fusion frame and g-frame.…”
We introduce the notion of a g-atomic subspace for a bounded linear operator and construct several useful resolutions of the identity operator on a Hilbert space using the theory of g-fusion frames. Also we shall describe the concept of frame operator for a pair of g-fusion Bessel sequences and some of their properties.
“…K-fusion frame. In [3], authors, introduced a generalization of fusion frame, K-fusion frame, and scrutinized the equivalence between atomic subspaces and K-fusion frames. Kfusion frame is used to reconstruct signals from range of a bounded linear operator K.…”
Section: Frame a Collection {Fmentioning
confidence: 99%
“…Frame theory literature became richer through several generalizations-fusion frame (frames of subspaces) [4,6] , G-frame (generalized frames) [22], K-frame (atomic systems) [14], K-fusion frame (atomic subspaces) [3], etc. and these generalizations have been proved to be useful in various applications.…”
K-fusion frames are generalizations of fusion frames in frame theory. This article characterizes various kinds of property of K-fusion frames. Several perturbation results on K-fusion frames are formulated and analyzed.
“…Frame theory literature became richer through several generalizations, namely, G-frame (generalized frames) [3], K-frame (frames for operators (atomic systems)) [4], fusion frame (frames of subspaces) ( [5,6]), K-fusion frame (atomic subspaces) [7], etc. and some spin-off applications by means of Gabor analysis in ( [8,9]), dynamical system in mathematical physics in [10], nature of shift invariant spaces on the Heisenberg group in [11], characterizations of discrete wavelet frames in C N in [12], extensions of dual wavelet frames in [13], constructions of disc wavelets in [14], orthogonality of frames on locally compact abelian groups in [15] and many more.…”
In a separable Hilbert space H, two frames {f i } i∈I and {g i } i∈I are said to be woven if there are constants 0 < A ≤ B so that for every σ ⊂ I,This article provides methods of constructing woven frames. In particular, bounded linear operators are used to create woven frames from a given frame.Several examples are discussed to validate the results. Moreover, the notion of woven frame sequences is introduced and characterized.
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