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.

Controlled g-fusion frame in Hilbert space

2021 ◽  
pp. 2150009
Author(s):  
Hanbing Liu ◽  
Yongdong Huang ◽  
Fengjuan Zhu
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Iterative Algorithms ◽  
Bessel Sequence ◽  
Numerical Efficiency ◽  
Fusion Frames ◽  
Controlled Fusion ◽  
Fusion Frame

Fusion frame is a generalization of frame, which can analyze signals by projecting them onto multidimensional subspaces. Controlled fusion frame as generalization of fusion frame, it can improve the numerical efficiency of iterative algorithms for inverting the fusion frame operators. In this paper, we first introduce the notion of controlled g-fusion frame, discuss several properties of controlled g-fusion Bessel sequence. Then, we present some sufficient conditions and some characterizations of controlled g-fusion frames. Finally, we study the sum of controlled g-fusion frames.

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 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.

Construction of g-fusion frames in Hilbert spaces

2020 ◽  
Vol 23 (02) ◽  
pp. 2050015
Author(s):  
Vahid Sadri ◽  
Gholamreza Rahimlou ◽  
Reza Ahmadi ◽  
Ramazan Zarghami Farfar
Keyword(s):  
Hilbert Spaces ◽  
Closed Range ◽  
Fusion Frames ◽  
Interesting Topic ◽  
Fusion Frame

After introducing g-frames and fusion frames by Sun and Casazza, respectively, combining these frames together is an interesting topic for research. In this paper, we introduce the generalized fusion frames or g-fusion frames for Hilbert spaces and give characterizations of these frames from the viewpoint of closed range and g-fusion frame sequences. Also, the canonical dual g-fusion frames are presented and we introduce a Parseval g-fusion frame.

FUSION FRAMES AND g-FRAMES IN HILBERT C*-MODULES

2008 ◽  
Vol 06 (03) ◽  
pp. 433-446 ◽  
Author(s):  
AMIR KHOSRAVI ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
Frame Theory ◽  
Fusion Frames ◽  
Perturbation Result ◽  
Fusion Frame

The notion of frame has some generalizations such as frames of subspaces, fusion frames and g-frames. In this paper, we introduce fusion frames and g-frames in Hilbert C*-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also generalize a perturbation result in frame theory to g-frames in Hilbert spaces. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity.

scholarly journals The Duals of Fusion Frames for Experimental Data Transmission Coding of High Energy Physics

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jinsong Leng ◽  
Qixun Guo ◽  
Tingzhu Huang
Keyword(s):  
Experimental Data ◽  
Data Transmission ◽  
High Energy Physics ◽  
Matrix Representation ◽  
High Energy ◽  
Fusion Frames ◽  
Fusion Frame ◽  
The Matrix ◽  
Frame System ◽  
Energy Physics

The experimental data transmission is an important part of high energy physics experiment. In this paper, we connect fusion frames with the experimental data transmission implement of high energy physics. And we research the utilization of fusion frames for data transmission coding which can enhance the transmission efficiency, robust against erasures, and so forth. For this application, we first characterize a class of alternate fusion frames which are duals of a given fusion frame in a Hilbert space. Then, we obtain the matrix representation of the fusion frame operator of a given fusion frame system in a finite-dimensional Hilbert space. By using the matrix representation, we provide an algorithm for constructing the dual fusion frame system with its local dual frames which can be used as data transmission coder in the high energy physics experiments. Finally, we present a simulation example of data coding to show the practicability and validity of our results.

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 On some general scheme of constructing iterative methods for searching the Nash equilibrium in concave games

2021 ◽  
Vol 13 (3) ◽  
pp. 75-121
Author(s):  
Андрей Владимирович Чернов ◽  
Andrey Chernov
Keyword(s):  
Nash Equilibrium ◽  
Iterative Process ◽  
Numerical Experiments ◽  
General Scheme ◽  
Iterative Algorithms ◽  
Specific Character ◽  
Residual Function ◽  
Arbitrary Continuous Function ◽  
Finite Dimensional ◽  
The Subject

The subject of the paper is finite-dimensional concave games id est noncooperative $n$-person games with objective functionals concave with respect to `their own' variables. For such games we investigate the problem of designing iterative algorithms for searching the Nash equilibrium with convergence guaranteed without requirements concerning objective functionals such as smoothness and as convexity in `strange' variables or another similar hypotheses (in the sense of weak convexity, quasiconvexity and so on). In fact, we prove some assertion analogous to the theorem on convergence of $M$-Fej\'{er iterative process for the case when an operator acts in a finite-dimensional compact and nearness to an objective set is measured with the help of arbitrary continuous function. Then, on the base of this assertion we generalize and develop the approach suggested by the author formerly to searching the Nash equilibrium in concave games. The last one can be regarded as "a cross between" the relaxation algorithm and the Hooke-Jeeves method of configurations (but taking into account a specific character of the the residual function being minimized). Moreover, we present results of numerical experiments with their discussion.

scholarly journals SENSITIVE VERSUS CLASSICAL SINGULAR PERTURBATION PROBLEM VIA FOURIER TRANSFORM

2006 ◽  
Vol 16 (11) ◽  
pp. 1783-1816 ◽  
Author(s):  
N. MEUNIER ◽  
E. SANCHEZ-PALENCIA
Keyword(s):  
Fourier Transform ◽  
Singular Perturbation ◽  
Elliptic Boundary ◽  
Characteristic Parameter ◽  
Elliptic Boundary Value ◽  
Perturbation Problem ◽  
Formal Asymptotics ◽  
Ill Posed ◽  
The Subject ◽  
Thin Shell Theory

We consider a class of singular perturbation elliptic boundary value problems depending on a parameter ε which are classical for ε > 0 but highly ill-posed for ε = 0 as the boundary condition does not satisfy the Shapiro–Lopatinskii condition. This kind of problems is motivated by certain situations in thin shell theory, but we only deal here with model problems and geometries allowing a Fourier transform treatment. We consider more general loadings and more singular perturbation terms than in previous works on the subject. The asymptotic process exhibits a complexification phenomenon: in some sense, the solution becomes more and more complicated as ε decreases, and the limit does not exist in classical distribution theory (it may only be described in spaces of analytical functionals not enjoying localization properties). This phenomenon is associated with the emergence of the new characteristic parameter |log ε|. Numerical experiments based on a formal asymptotics are presented, exhibiting features which are unusual in classical elliptic equations theory. We also give a Fourier transform treatment of classical singular perturbations in order to exhibit the drastic differences with the present situation.

scholarly journals Frames and subspaces

2018 ◽  
Author(s):  
◽  
Desai Cheng
Keyword(s):  
Phase Retrieval ◽  
Wasserstein Distance ◽  
Complex Case ◽  
Exact Constant ◽  
Frame Operator ◽  
Parseval Frame ◽  
Fusion Frames ◽  
Open Questions ◽  
Rank 2 ◽  
Equiangular Lines

This thesis will consist of three parts. In the first part we find the closest probabilistic Parseval frame to a given probabilistic frame in the 2 Wasserstein Distance. It is known that in the traditional [symbol]2 distance the closest Parseval frame to a frame [phi] = {[symbol]i} N[i=1] [symbol] R[d] is [phi[�] = {[symbol] � i }N[i]=1 = {S [-1/2][[symbol]i]} N[i=1] where S is the frame operator of [phi]. We use this fact to prove a similar statement about probabilistic frames in the 2 Wasserstein metric. In the second part, we will associate a complex vector with a rank 2 real projection. Using this association we will answer many open questions in frame theory. In particular we will prove Edidin's theorem in phase retrieval in the complex case, answer a question on mutually unbiased bases, a question on equiangular lines, and a question on fusion frames. In the last part we will give a way to calculate the exact constant for the [symbol]1 � [symbol]2 inequality and use this method to prove a couple of interesting theorems

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