G-frames In Hilbert Spaces Research Articles

Redundancy, weaving and $Q$-dual of $K$-g-frames in Hilbert spaces

In this paper we study exact $K$-g-frames, weaving of $K$-g-frames and $Q$-duals of g-frames in Hilbert spaces. We present a sufficient condition for a g-Bessel sequence to be an exact $K$-g-frame. Given two woven pairs $(\Lambda, \Gamma)$ and $(\Theta, \Delta)$ of $K$-g-frames, we investigate under what conditions $\Lambda$ can be $K$-g-woven with $\Delta$ if $\Gamma$ is $K$-g-woven with $\Theta$. Given a $K$-g-frame $\Lambda$ and its dual $\Gamma$ on $\mathcal{U}$, we construct a new pair based on $\Lambda$ and $\Gamma$ so that they are woven on a subspace $R(K)$ of $\mathcal{U}$. Finally, we characterize the $Q$-dual of g-frames using their induced sequences.

Characteizations of woven g-frames and weaving g-frames in Hilbert spaces and C*-modules

Characteizations of woven g-frames and weaving g-frames in Hilbert spaces and C*-modules

Controlled g-frames and dual g-frames in Hilbert spaces

As generalizations of g-frames and controlled frames, the theory of controlled g-frames has been deeply studied. This paper addresses the controlled g-frames and dual g-frames in Hilbert spaces. We first present some equivalent characterizations of controlled g-frames. Then, we introduce the concepts of controlled dual g-frames and controlled dual g-frames operator, get some properties of them. Finally, we obtain some characterizations of the controlled dual g-frames for a given controlled g-frame by the method of operator theory.

Q-duals and Q-approximate duals of g-frames in Hilbert spaces

In this paper we mainly discuss the properties of Q-duals and Q-approximate duals of g-frames in Hilbert spaces. Given and being a pair of Q-dual, being some kind of perturbed sequence of in general is not a Q-approximate dual of We then give four different kinds of perturbed conditions such that and a perturbed sequence of are possible to be a pair of Q-approximate dual. We also provide several different methods to construct Q-duals and Q-approximate duals of g-frames. Finally, we give two equivalent characterizations of Q-duals and Q-approximate duals by using the associated induced sequences.

Characterizations and redundancies of g-frames in Hilbert spaces

Let { Λ j : j ∈ J } be a g-Bessel sequence of a Hilbert space. This study uses the type-II induced sequences { Γ j k Λ j : j ∈ J , k ∈ K j } of { Λ j : j ∈ J } to characterize { Λ j : j ∈ J } to be a tight g-frame and a g-orthonormal basis. We also obtain the exact relationship between the synthesis operators of { Λ j : j ∈ J } and its type-II induced sequences. We then use the type-I induced sequences of { Λ j : j ∈ J } to characterize the (maximum) robustness to erasures and (maximum) uniform excess of { Λ j : j ∈ J } , and vice versa. Finally, we estimate the upper error bound of type-I induced sequences under some perturbations of { Λ j : j ∈ J } and { Γ j : j ∈ J } , and vice versa.

Some results about g-frames in Hilbert spaces

The concept of g-frame is a natural extension of the frame. This article mainly discusses the relationship between some special bounded linear operators and g-frames, and characterizes the properties of g-frames. In addition, according to the operator spectrum theory, the eigenvalues are introduced into the g-frame theory, and a new expression of the best frame boundary of the g-frame is given.

Some new results on the weaving of K-g-frames in Hilbert spaces

Abstract In this paper, we provide some conditions for a K-woven pair of K-g-frames to be preserved under an operator and particularly, we report that applying two different operators to a K-woven pair of K-g-frames can leave them K-woven. Several new methods on the construction of K-woven pair of K-g-frames are also given. We end the paper with a new perturbation result on the weaving of K-g-frames, which shows that, under the perturbation condition involved in one known result on this topic, two K-g-frames can be K-woven in the whole space, not merely in the subspace Range(K).

On Optimal Estimate of the Block Orthogonal Greedy Algorithm for g-Frames

We obtain an optimal approximation of the block orthogonal greedy algorithm with regard to g-frames in Hilbert spaces and prove an optimal approximation result analogous to the result obtained by E. D. Livshitz [15] using dictionaries.

On Weaving g-Frames for Hilbert Spaces

Weaving frames are powerful tools in wireless sensor networks and pre-processing signals. In this paper, we introduce the concept of weaving for g-frames in Hilbert spaces. We first give some properties of weaving g-frames and present two necessary conditions in terms of frame bounds for weaving g-frames. Then we study the properties of weakly woven g-frames and give a sufficient condition for weaving g-frames. It is shown that weakly woven is equivalent to woven. Two sufficient conditions for weaving g-Riesz bases are given. And a weaving equivalent of an unconditional g-basis for weaving g-Riesz bases is considered. Finally, we present Paley-Wiener-type perturbation results for weaving g-frames.

Operators on controlled $K$-g-frames in Hilbert spaces

Operators on controlled $K$-g-frames in Hilbert spaces

Operator characterizations and constructions of continuous g-frames in Hilbert spaces

ABSTRACT This paper addresses continuous g-frames which are extensions of g-frames and continuous frames. First, we study the dual of continuous g-frames, which are critical components in reconstructions. We give the characterizations of dual continuous g-frames in terms of continuous g-preframe operators. The formulae of dual continuous g-frames for a given continuous g-frame are given. Second, we discuss the constructions of continuous g-frames in Hilbert spaces. We mainly consider the constructions of new continuous g-frames from known ones under certain conditions, which generalize the corresponding results on g-frames. In particular, we obtain a necessary and sufficient condition for the finite sum of continuous g-frames to be a continuous g-frame.

Characterization and stability of approximately dual g-frames in Hilbert spaces

This paper addresses approximately dual g-frames. First, we establish a connection between approximately dual g-frames and dual g-frames and obtain a characterization of approximately dual g-frames. Second, we give results on stability of approximately dual g-frames, which cover the results obtained by other authors.

On sum and stability of g-frames in Hilbert spaces

In this paper, we give some new results on sum and stability of g-frames in Hilbert spaces. Since the finite sum of g-frames may not be a g-frame for the Hilbert space, we give a necessary and sufficient condition and some sufficient conditions for the finite sum of g-frames to be a g-frame. We also show that every g-sequence in Hilbert space can be expanded to a tight g-frame by adding a linear bounded operator. Moreover, we obtain some sufficient conditions under which g-frames (and the finite sum of g-frames) are stable under small perturbations.

Duality and biorthogonality for g-frames in Hilbert spaces

Duality and biorthogonality for g-frames in Hilbert spaces

Approximate duals and nearly Parseval frames

In this paper we introduce approximate duality of g-frames in Hilbert $C^\ast$-modules and we show that approximate duals of g-frames in Hilbert $C^\ast$-modules share many useful properties with those in Hilbert spaces. Moreover, we obtain some new results for approximate duality of frames and g-frames in Hilbert spaces; in particular, we consider approximate duals of $\varepsilon$-nearly Parseval and $\varepsilon$-close frames.

Approximate duality of g-frames in Hilbert spaces

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.

Some equalities and inequalities for g-Bessel sequences in Hilbert spaces (II)

G-frames, which include many generalizations of frames such as frames of subspaces or fusion frames, oblique frames, and pseudo-frames, are natural generalizations of frames in Hilbert spaces. They have some properties similar to those of frames in Hilbert spaces, but not all of their properties are similar. For example, exact frames are equivalent to Riesz bases, but exact g-frames are not equivalent to g-Riesz bases. Some authors have extended the equalities and inequalities for frames and dual frames to g-frames and dual g-frames in Hilbert spaces. In this paper, we establish some new equalities and inequalities for g-Bessel sequences or g-frames in Hilbert spaces. We also give a necessary and sufficient condition that the equality occurs in one of these inequalities. Our results generalize and improve the remarkable results which had been obtained by Balan, Casazza and Gavruta.

Perturbations of Invertible Operators and Stability of g-Frames in Hilbert Spaces

In this paper, we study the perturbations of invertible operators and stability of g-frames in Hilbert spaces. In particular, we obtain some conditions under which the perturbations of an invertible operator are still an invertible operator, the perturbations of a right invertible operator or a surjective operator are still a right invertible operator or surjective operator. Then we apply the perturbations of invertible operators to study the stability of g-frames which is close related with the invertibility (or right invertibility) property of operators.

Equivalency Relations between Continuous g-Frames and Stability of Alternate Duals of Continuous g-Frames in Hilbert -Modules

We introduce the modular continuous g-Riesz basis to improve one existing result for continuous g-Riesz basis in Hilbert -modules, and then we study the equivalency relations between continuous g-frames in Hilbert -modules, and, in particular, we obtain two necessary and sufficient conditions under which two continuous g-frames are similar. Finally, we generalize a stability result for alternate duals of g-frames in Hilbert spaces to alternate duals of continuous g-frames in Hilbert -modules.

Fusion frames and G-frames in Banach spaces

Fusion frames and g-frames in Hilbert spaces are generalizations of frames, and frames were extended to Banach spaces. In this article we introduce fusion frames, g-frames, Banach g-frames in Banach spaces and we show that they share many useful properties with their corresponding notions in Hilbert spaces. We also show that g-frames, fusion frames and Banach g-frames are stable under small perturbations and invertible operators.