scholarly journals Controlled G-Frames and Their G-Multipliers in Hilbert spaces

2013 ◽  
Vol 21 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Asghar Rahimi ◽  
Abolhassan Fereydooni
Keyword(s):  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Multiplier Operator ◽  
Frame Operator ◽  
Fixed Sequence ◽  
Numerical Efficiency ◽  
Bessel Sequences ◽  
Almost All

Abstract Multipliers have been recently introduced by P. Balazs as operators for Bessel sequences and frames in Hilbert spaces. These are opera- tors that combine (frame-like) analysis, a multiplication with a fixed sequence ( called the symbol) and synthesis. One of the last extensions of frames is weighted and controlled frames that introduced by P.Balazs, J-P. Antoine and A. Grybos to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Also g-frames are the most popular generalization of frames that include almost all of the frame extensions. In this manuscript the concept of the controlled g- frames will be defined and we will show that controlled g-frames are equivalent to g-frames and so the controlled operators C and C' can be used as preconditions in applications. Also the multiplier operator for this family of operators will be introduced and some of its properties will be shown.

scholarly journals Controlled generalized fusion frame in the tensor product of Hilbert spaces

2021 ◽  
pp. 1-18
Author(s):  
Prasenjit Ghosh ◽  
Tapas Kumar Samanta
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Frame Operator ◽  
Fusion Frame ◽  
Bessel Sequences

We present controlled by operators generalized fusion frame in the tensor product of Hilbert spaces and discuss some of its properties. We also describe the frame operator for a pair of controlled $g$-fusion Bessel sequences in the tensor product of Hilbert spaces.

scholarly journals WEIGHTED AND CONTROLLED FRAMES: MUTUAL RELATIONSHIP AND FIRST NUMERICAL PROPERTIES

2010 ◽  
Vol 08 (01) ◽  
pp. 109-132 ◽  
Author(s):  
PETER BALAZS ◽  
JEAN-PIERRE ANTOINE ◽  
ANNA GRYBOŚ
Keyword(s):  
Iterative Algorithms ◽  
Gabor Frames ◽  
Dual Frame ◽  
Mutual Relationship ◽  
Frame Operator ◽  
Numerical Efficiency ◽  
Frame Condition ◽  
Frame Bound ◽  
Special Case ◽  
Numerical Advantage

Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we develop systematically these notions, including their mutual relationship. We will show that controlled frames are equivalent to standard frames and so this concept gives a generalized way to check the frame condition, while offering a numerical advantage in the sense of preconditioning. Next, we investigate weighted frames, in particular their relation to controlled frames. We consider the special case of semi-normalized weights, where the concepts of weighted frames and standard frames are interchangeable. We also make the connection with frame multipliers. Finally, we analyze weighted frames numerically. First, we investigate three possibilities for finding weights in order to tighten a given frame, i.e. decrease the frame bound ratio. Then, we examine Gabor frames and how well the canonical dual of a weighted frame is approximated by the inversely weighted dual frame.

scholarly journals Controlled K−g−Fusion Frames in Hilbert C∗−Modules

2022 ◽  
Vol 20 ◽  
pp. 1
Author(s):  
Mohamed Rossafi ◽  
Fakhr-dine Nhari
Keyword(s):  
Iterative Algorithms ◽  
Frame Operator ◽  
Numerical Efficiency ◽  
Fusion Frames ◽  
Fusion Frame ◽  
Perturbation Problem ◽  
The Subject

Controlled frames have been the subject of interest because of its ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the concepts of controlled g−fusion frame and controlled K−g−fusion frame in Hilbert C∗−modules and we give some properties. Also, we study the perturbation problem of controlled K−g−fusion frame. Moreover, an illustrative example is presented to support the obtained results.

scholarly journals A New Inequality for Frames in Hilbert Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Zhong-Qi Xiang
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Bessel Sequences

We obtain a new inequality for frames in Hilbert spaces associated with a scalar and a bounded linear operator induced by two Bessel sequences. It turns out that the corresponding results due to Balan et al. and Găvruţa can be deduced from our result.

scholarly journals Iterative methods for generalized split feasibility problems in Banach spaces

2017 ◽  
Vol 33 (1) ◽  
pp. 09-26
Author(s):  
QAMRUL HASAN ANSARI ◽  
◽  
AISHA REHAN ◽  
◽  
Keyword(s):  
Weak Convergence ◽  
Banach Spaces ◽  
Iterative Methods ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Algorithms ◽  
Approximate Solutions ◽  
Split Feasibility Problems ◽  
Convergence Results ◽  
Split Feasibility

Inspired by the recent work of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 23 (2015), 205–221], in this paper, we study generalized split feasibility problems (GSFPs) in the setting of Banach spaces. We propose iterative algorithms to compute the approximate solutions of such problems. The weak convergence of the sequence generated by the proposed algorithms is studied. As applications, we derive some algorithms and convergence results for some problems from nonlinear analysis, namely, split feasibility problems, equilibrium problems, etc. Our results generalize several known results in the literature including the results of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, SetValued Var. Anal., 23 (2015), 205–221].

scholarly journals Algorithm for Solving a New System of Generalized Nonlinear Quasi-Variational-Like Inclusions in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shamshad Husain ◽  
Sanjeev Gupta ◽  
Huma Sahper
Keyword(s):  
Exact Solutions ◽  
Existence Theorem ◽  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Approximate Solutions ◽  
Cocoercive Operator ◽  
New System

We introduce and study a new system of generalized nonlinear quasi-variational-like inclusions with H(·,·)-cocoercive operator in Hilbert spaces. We suggest and analyze a class of iterative algorithms for solving the system of generalized nonlinear quasi-variational-like inclusions. An existence theorem of solutions for the system of generalized nonlinear quasi-variational-like inclusions is proved under suitable assumptions which show that the approximate solutions obtained by proposed algorithms converge to the exact solutions.

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