Prasenjit Ghosh, T. K. Samanta
The concepts of controlled frames and it's dual in n-Hilbert spaces and their tensor products have been introduced and then some of their characterizations are given. We further study the relationship between controlled frame and bounded linear operator in tensor product of n-Hilbert spaces. At the end, the direct sum of controlled frames in n-Hilbert space is being considered.
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In this paper, we introduce and we study the concept of Continuous Controlled K-Frame for Hilbert $C^{\ast}$-Modules wich are generalizations of discrete Controlled K-Frames.
We introduce the notion of continuous controlled g-fusion frame in Hilbert space which is the generalization of discrete controlled g-fusion frame and give an example. Some characterizations of continuous controlled g-fusion frame have been presented. We define the frame operator and multiplier of continuous controlled g-fusion Bessel families in Hilbert spaces. Continuous resolution of the identity operator on a Hilbert space using the theory of continuous controlled g-fusion frame is being considered. Finally, we discuss perturbation results of continuous controlled g-fusion frame.
The notion of controlled frames for Hilbert spaces were introduced by Balazs, Antoine and Grybos to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Controlled Frame Theory has a great revolution in recent years. This Theory have been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. In this paper we introduce and study the extension of this notion to integral frame for Hilbert $C^{\ast}$-module. Also we give some characterizations between integral frame in Hilbert $C^{\ast}$-module
Frame Theory has a great revolution for recent years. This Theory has been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. The purpose of this paper is the introduction and the study of the new concept that of Continuous Controlled K-g-Frame for Hilbert $C^{\ast}$-Modules wich is a generalizations of discrete Controlled K-g-Frames in Hilbert $C^{\ast}$-Modules. Also we establish some results.
In this paper, we characterize and study the concept of controlled continuous $g$-frame which is an extension of continuous $g$-frame in Hilbert spaces. We introduce the concept of controlled continuous dual $g$-frame and observe some interesting properties of it. Finally, we characterize all controlled continuous dual $g$-frames of a controlled continuous $g$-frame.
In this paper, we first introduce the notion of controlled weaving K-g-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving K-g-frames in separable Hilbert spaces. Also, a characterization of controlled weaving K-g-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled K-g-woven frames gives ordinary weaving K-frames and vice-versa.
To improve the numerical efficiency of iterative algorithms for inverting the frame operator, the controlled frame was introduced by Balazs et al. \cite{Balazs}, and has since been given more importance. In this paper, we introduce the concept of controlled g-frames in Hilbert $C^{*}$-modules. We establish the equivalent condition for controlled $g$-frame using operator theoretic approach. We investigate some operator theoretic characterizations of controlled $g$-frames and controlled $g$-Bessel sequences. We also established the relationship between $g$-frames and controlled $g$-frames in Hil…
The frame theory is dynamic and exciting with various pure and applied mathematics applications. In this paper, we introduce and study the concept of Controlled Continuous $\ast$-$g$-Frames in Hilbert $C^{\ast}$-Modules, which is a generalization of discrete controlled $\ast$-$g$-Frames in Hilbert $C^{\ast}$-Modules. Also, we give some properties.
Controlled g-atomic subspace for a bounded linear operator is being presented and a characterization has been given. We give an example of controlled K-g-fusion frame. We construct a new controlled K-g-fusion frame for the Hilbert space H ? X using the controlled K-g-fusion frames of the Hilbert spaces H and X. Several useful resolutions of the identity operator on a Hilbert space using the theory of controlled g-fusion frames have been discussed. Frame operator for a pair of controlled g-fusion Bessel sequences has been introduced.
Controlled frames and g-frames were considered recently as generalizations of frames in Hilbert spaces. In this paper we generalize some of the known results in frame theory to controlled g-frames. We obtain some new properties of controlled g-frames and obtain new controlled g-frames by considering controlled g-frames for its components. And we obtain some new resolutions of the identity. Furthermore, we study the stabilities of controlled g-frames under small perturbations.
K-frames were recently introduced by L. Gǎvruta in Hilbert spaces to study atomic systems with respect to bounded linear operator. Also controlled frames have been recently introduced by P. Balazs in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this manuscript, we will define the concept of the controlled K-frames and will show that controlled K-frames are equivalent to K-frames and so the controlled operator C can be used as preconditions in applications.
Controlled $E$-frames in Hilbert spaces
本文提出Hilbert空间中受控$E$-框架的概念，研究其基本性质，并完整刻画了给定受控$E$-框架的所有受控$E$-对偶。
Construction of continuous controlled K-g-fusion frames in Hilbert spaces
本文提出了Hilbert空间中连续受控K-g-融合框架的概念，作为离散框架的推广，研究了其刻画、与商算子的关系及稳定性。
Weaving continuous controlled K-g-fusion frames in Hilbert spaces
该研究在希尔伯特空间中提出编织连续受控K-g-融合框架概念，给出其若干刻画，推广了两类已有框架的相关结果，并研究了该框架的扰动性质。
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 notion of controlled $K$-frame in Hilbert $C^{*}$-modules. We establish the equivalent condition for controlled $K$-frame. We investigate some operator theoretic characterizations of controlled $K$-frames and controlled Bessel sequences. Moreover we establish the relationship between the $K$-frames and controlled $K$-frames. We also investigate the invariance of a $C$-controlled $K$-frame under a…
Multipliers have been recently introduced by P. Balazs as operators for Bessel sequences and frames in Hilbert spaces. These are operators that combine (frame-like) analysis, a multiplication with a fixed sequence (called the symbol) and synthesis. Weighted and controlled frames have been introduced 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 c…
On $E$-frames in Hilbert spaces
本文研究希尔伯特空间中的E-框架，提出受控E-框架概念，并完整刻画了给定受控E-框架的所有受控E-对偶。
In this paper, we introduce controlled frames in Hilbert $C^*$-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. Next, we give a characterization of controlled frames in Hilbert $C^*$-module. Also multiplier operators for controlled frames in Hilbert $C^*$-modules will be defined and some of its properties will be shown. Finally, we investigate weighted frames in Hilbert $C^*$-modules and verify their relations to controlled frames and multiplier operators.
Frame Theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. In this paper, we introduce the concept of Controlled K-operator frame for the space $End_{\mathcal{A}}^{\ast}(\mathcal{H})$ of all adjointable operators on a Hilbert $\mathcal{A}$-module $\mathcal{H}$ and we establish some results.
Trace-Controlled Operator Selection and Discretization of Continuous Frames in Hilbert \(C^*\)-Modules
本文针对有限维$C^*$-代数上的Hilbert $C^*$-模，提出迹控制算子选择定理，并应用于连续框架的离散化，得到具有显式框架界和受控分离的采样族。
Frame Theory has a great revolution in recent years, this Theory have been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. The purpose of this paper is the introduction and the study of the concept of Controlled Continuous g-Frames in Hilbert $C^{\ast}$-Modules. Also we give some properties.
Frame Theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. In this paper, we introduce the concept of Controlled $\ast$-$K$-operator frame for the space $End_{\mathcal{A}}^{\ast}(\mathcal{H})$ of all adjointable operators on a Hilbert $\mathcal{A}$-module $\mathcal{H}$ and we establish some results.
$K$-fusion frames are a generalization of fusion frames in frame theory. In this paper, we extend the concept of controlled fusion frames to controlled $K$-fusion frames, and we develop some results on the controlled $K$-fusion frames for Hilbert spaces, which generalized some well known of controlled fusion frames case. also we discuss some characterizations of controlled Bessel $K$-fusion sequences and of controlled Bessel $K$-fusion. Further, we analyse stability conditions of controlled $K$-fusion frames under perturbation.
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 pertubation problem of controlled K-g-fusion frame. Moreover, an illustrative examples is presented to support the obtained results.
A Hilbert space frame on $R^n$ is {\it scalable} if we can scale the vectors to make them a tight frame. There are known classifications of scalable frames. There are two basic questions here which have never been answered in any $R^n$: Given a frame in $R^n$, how do we scale the vectors to minimize the condition number of the frame? I.e. How do we scale the frame to make it as tight as possible? If we are only allowed to use scaling numbers from the interval $[1-ε,1+ε]$, how do we scale the frame to minimize the condition number? We will answer these two questions in $R^2$ to begin the proces…
Controlled frames in Hilbert spaces have been introduced by Balazs, Antoine and Grybos to improve the numerical output of in relation to algorithms for inverting the frame operator. In this paper we have introduced and displayed some new concepts and results on controlled fusion frames for Hilbert spaces. It is shown that controlled fusion frames as a generalization of fusion frames give a generalized way to obtain numerical advantage in the sense of reconditioning to check the fusion frame condition. For this end, we introduce the notion of Q-duality for Controlled fusion frames. Also, we sur…
K-frames were recently introduced by L. Gǎvruta in Hilbert spaces to study atomic systems with respect to bounded linear operator. Also controlled frames have been recently introduced by Balazs, Antoine and Grybos in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this manuscript, the concept of controlled K-frames will be studied and the stability of Controlled K-frames under compact perturbation will be discussed.
Controlled finite continuous frames
该研究在有限维希尔伯特空间中提出受控有限连续框架与Parseval受控积分框架，研究其性质并刻画了构造受控积分有限框架的算子。
Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert-Schmidt or a trace-class operator. Similar to standar…
Constructions of bi-g-fusion frame in Hilbert space
该研究提出了希尔伯特空间中作为受控g-fusion框架推广的bi-g-fusion框架概念，给出示例并探讨了其在希尔伯特空间张量积中的情况。
In this note, following the complex theory, we examine discrete controlled frames, discrete weighted frames and frame multipliers in a non-commutative setting, namely in a left quaternionic Hilbert space. In particular, we show that the controlled frames are equivalent to usual frames under certain conditions. We also study connection between frame multipliers and weighted frames in the same Hilbert space.
On perturbation of Hilbert-Schmidt frames
本文研究了在结构修改下Hilbert-Schmidt框架的扰动问题，建立了扰动后序列保持Hilbert-Schmidt框架性质的显式条件，讨论了有限和无限情况下的稳定性界及应用示例。
In this paper we examine the general theory of continuous frame multipliers in Hilbert space. These operators are a generalization of the widely used notion of (discrete) frame multipliers. Well-known examples include Anti-Wick operators, STFT multipliers or Calderón- Toeplitz operators. Due to the possible peculiarities of the underlying measure spaces, continuous frames do not behave quite as well as their discrete counterparts. Nonetheless, many results similar to the discrete case are proven for continuous frame multipliers as well, for instance compactness and Schatten class properties. F…
In this paper, we introduce and study the frames in separable quaternionic Hilbert spaces. Results on the existence of frames in quaternionic Hilbert spaces have been given. Also, a characterization of frame in quaternionic Hilbert spaces in terms of frame operator is given. Finally, a Paley-Wiener type perturbation result for frames in quaternionic Hilbert space has been obtained.
Frame Theory has a great revolution in recent years. This Theory have been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. In this paper we consider the stability of continuous operator frame and continuous $K$-operator frames in Hilbert $C^{\ast}$-Modules under perturbation and we establish some properties
In this paper, we introduce and study frame of operators in quaternionic Hilbert spaces as a generalization of g frames which in turn generalized various notions like Pseduo frames, bounded quasi-projectors and frame of subspaces (fusion frames) in separable quaternionic Hilbert spaces.
In this paper we introduced the concept of continuous relay fusion frames in Hilbert spaces. And we define the dual frames for continuous relay fusion frames. Finally we study the perturbation probleme of continuous relay fusion frames.
In this paper, our aim is to introduce the concept of a frame in n-Hilbert space and describe some of their properties. We further discuss tight frame relative to n-Hilbert space. At the end, we study the relationship between frame and bounded linear operator in n-Hilbert space.
We characterize the frames on an infinite dimensional separable Hilbert space that can be projected to a tight frame for an infinite dimensional subspace. A result of Casazza and Leon states that an arbitrary frame for a 2N- or (2N-1)-dimensional Hilbert space can be projected to a tight frame for an N-dimensional subspace. Surprisingly, we demonstrate a large class of frames for infinite dimensional Hilbert spaces which cannot be projected to a tight frame for any infinite dimensional subspace.
In this paper we have some new results on sums of Hilbert space frames and Riesz bases. We also have a correction for some results in "S. Obeidat et al., Sums of Hilbert space frames, J. Math. Anal. Appl. 351 (2009) 579-585."
Frames in a separable quaternionic Hilbert space were introduced and studied in [17] to have more applications. In this paper, we extend the study of frames in quaternionic Hilbert spaces and introduce different types of duals of a frame in separable quaternionic Hilbert spaces. As an application, we give the orthogonal projection of $\ell^2(\HH)$ onto the range of analysis operator of the given frame, in terms of elements of canonical dual frame and elements of the frame in quaternionic Hilbert space. Finally, we give an expression for the orthogonal projection in terms of operators related t…
Generalized fusion frame and some of their properties in tensor product of Hilbert spaces are described. Also, the canonical dual g-fusion frame in tensor product of Hilbert spaces is considered. Finally, the frame operator for a pair of g-fusion Bessel sequences in tensor product of Hilbert spaces is presented.
The notion of a K-frame in n-Hilbert space is presented and some of their characterizations are given. We verify that sum of two K-frames is also a K-frame in n-Hilbert space. Also, the concept of tight K-frame in n-Hilbert space is described and some properties of its are going to be established.
Generalized fusion frame in Quaternionic Hilbert spaces
本文研究四元数 Hilbert 空间中的广义融合框架，给出其基于框架算子的刻画，并利用可逆有界右 Q-线性算子构造 g-fusion frame。
Introduction of frame in tensor product of n-Hilbert spaces
本研究针对n-希尔伯特空间张量积仍为n-希尔伯特空间的特性，引入框架概念，推广基的相关结果，探究框架与有界线性算子的关系，并讨论对偶框架，拓展了n-希尔伯特空间框架理论的研究范畴。
In the present paper the notion of continuous frames is introduced and some results of these frames are proved. Next, we give the concept of duals of continuous frames in Hilbert C*-modules and investigate some properties of them.
Continuous frames in n-Hilbert spaces and their tensor products
该研究提出n-希尔伯特空间中连续框架的概念，将其推广至n-希尔伯特空间的张量积场景，并进一步研究了对应空间及张量积下的对偶连续框架与连续Bessel乘子。
Kaczmarz algorithm in Hilbert space and tight frames
证明了希尔伯特空间中的任何紧框架都可以用Kaczmarz算法得到。给出了构造这种对应关系的一种显式方法。确定对应的唯一性。
In this paper, we study the Hilbert$-$Schmidt frame (HS-frame) theory for separable Hilbert spaces. We first present some characterizations of HS-frames and prove that HS-frames share many important properties with frames. Then, we show how the inverse of the HS-frame operator can be approximated using finite-dimensional methods. Finally, we present a classical perturbation result and prove that HS-frames are stable under small perturbations.
In this manuscript, the concept of dual and approximate dual for continuous frames in Hilbert spaces will be introduced. Some of its properties will be studied. Also, the relations between two continuous Riesz bases in Hilbert spaces will be clarified through examples.
Frame theory is recently an active research area in mathematics, computer science and engineering with many exciting applications in a variety of different fields. This theory has been generalized rapidly and various generalizations of frames in Hilbert spaces. In this papers we study the notion of dual continuous $K$-frames in Hilbert spaces. Also we etablish some new properties.
Frame theory has been rapidly generalized and various generalizations have been developed. In this paper, we present a brief survey of the frames in Hilbert $C^{\ast}$-modules, including frames, $\ast$-frames, g-frames, $\ast$-g-frames, $\ast$-$K$-g-frame, operator frame and $\ast$-$K$-operator in Hilbert $C^{\ast}$-modules. Various proofs are given for some results. Also, we prove some new results. Moreover, non-trivial examples are presented.
Local theory of frames and Schauder bases for Hilbert space
我们发展了有限维希尔伯特空间上框架的局部理论。特别地，有限维希尔伯特空间上的有界框架包含一个子集，该子集是空间的一个百分比（任意接近1）的良好Riesz基。我们还构造了一个希尔伯特空间的规范化框架，该框架包含H的Schauder基子集，但不包含H的Riesz基子集。
Nagy and Pollard-Hilding-Type Stability of Hilbert-Schmidt Frames
本文为可分Hilbert空间中Hilbert-Schmidt框架的扰动稳定性建立了基于Nagy型和Pollard-Hilding型不等式的充分条件，并提供了框架界的显式估计及示例。
Fusion frame and its alternative dual in tensor product of Hilbert spaces
该研究聚焦希尔伯特空间张量积中的融合框架，分析其相关性质与恒等算子的单位分解，并提出该场景下融合框架的替代对偶，丰富了泛函分析中融合框架的理论体系。
After introducing g-frames and fusion frames by Sun and Casazza, 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 Parseval g-fusion frames.
This is a short introduction to Hilbert space frame theory and its applications for those outside the area who want to enter the subject. We will emphasize finite frame theory since it is the easiest way to get into the subject.
Pairs of Woven continuous frames in Hilbert spaces
该研究引入希尔伯特空间的连续交织框架概念，提出构造交织连续框架对的新方法，推导其与对偶框架相关的新性质，并给出基于微小扰动的构造途径，丰富了连续框架理论。
In this paper, we focus on frames of operators or K-frames on Hilbert spaces in Parseval cases. Since equal-norm tight frames play important roles for robust data transmission, we aim to study this topics on Parseval K-frames. We will show that each finite set of equal-norms of vectors can be extended to an equal-norm K-frame. Also, we present a correspondence between Parseval K-frames and the set of all closed subspaces of a finite Hilbert space. Finally, a construction of equal-norm dual K-frames will be introduced.
Naimark complements for Hilbert space Parseval frames are one of the most fundamental and useful results in the field of frame theory. We will show that actually all Hilbert space frames have Naimark complements which possess all the usual properties for Naimark complements with one notable exception. So these complements can be used for equiangular frames, RIP property, fusion frames etc. Along the way, we will correct a mistake in a recent fusion frame paper where chordal distances for Naimark complements are computed incorrectly.
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