scholarly journals Some New Characterizations andg-Minimality and Stability ofg-Bases in the Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xunxiang Guo
Keyword(s):  
Hilbert Spaces ◽  
Frame Theory ◽  
Schauder Basis ◽  
Simple Condition ◽  
Similar Role ◽  
Perturbation Result

The concept ofg-basis in the Hilbert spaces is introduced by Guo (2012) who generalizes the Schauder basis in the Hilbert spaces.g-basis plays the similar role ing-frame theory to that the Schauder basis plays in frame theory. In this paper, we establish some important properties ofg-bases in the Hilbert spaces. In particular, we obtain a simple condition under which some important properties established in Guo (2012) are still true. With these conditions, we also establish some new interesting properties ofg-bases which are related tog-minimality. Finally, we obtain a perturbation result aboutg-bases.

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 Approximative K-atomic decompositions and frames in Banach spaces

2018 ◽  
Vol 26 (1/2) ◽  
pp. 153-166
Author(s):  
Shah Jahan
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Atomic Decomposition ◽  
Necessary Conditions ◽  
Bounded Operator ◽  
Special Kind ◽  
Frame Theory ◽  
Bessel Sequence ◽  
Atomic Decompositions ◽  
Counter Example

L. Gǎvruţa (2012) introduced a special kind of frames, named K-frames, where K is an operator, in Hilbert spaces, which is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative K-atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative K-atomic decompositions in Banach spaces. Also some results on the existence of approximative K-atomic decompositions are obtained. We discuss several methods to construct approximative K-atomic decomposition for Banach Spaces. Further, approximative d-frame and approximative d-Bessel sequence are introduced and studied. Two necessary conditions are given under which an approximative d-Bessel sequence and approximative d-frame give rise to a bounded operator with respect to which there is an approximative K-atomic decomposition. Example and counter example are provided to support our concept. Finally, a possible application is given.

scholarly journals Some results about g-frames in Hilbert spaces

2022 ◽  
Vol 355 ◽  
pp. 02001
Author(s):  
Lan Luo ◽  
Jingsong Leng ◽  
Tingting Xie
Keyword(s):  
Hilbert Spaces ◽  
Natural Extension ◽  
Linear Operators ◽  
Frame Theory ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Operator Spectrum ◽  
The Relationship

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.

g-FRAMES AND MODULAR RIESZ BASES IN HILBERT C*-MODULES

2012 ◽  
Vol 10 (02) ◽  
pp. 1250013 ◽  
Author(s):  
AMIR KHOSRAVI ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Hilbert Spaces ◽  
Riesz Basis ◽  
Riesz Bases ◽  
Parseval Frame ◽  
C Algebra ◽  
Perturbation Result ◽  
Natural Way

In this paper we introduce modular Riesz basis, modular g-Riesz basis in Hilbert C*-modules in a very natural way and we show that they share many properties with Riesz basis and g-Riesz basis in Hilbert spaces. We also found that by using the fact that every finitely or countably generated Hilbert C*-module over a unital C*-algebra has a standard Parseval frame, we characterize g-frames, modular Riesz bases and modular g-Riesz bases. Finally we obtain a perturbation result for modular g-Riesz bases.

scholarly journals On a Generalization of the Hilbert Frame Generated by the Bilinear Mapping

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Migdad Ismailov ◽  
Fatima Guliyeva ◽  
Yusif Nasibov
Keyword(s):  
Hilbert Spaces ◽  
Frame Theory ◽  
Frame Operator ◽  
Bilinear Mapping ◽  
Surjective Operator ◽  
The Relationship

The concept ofb-frame which is a generalization of the frame in Hilbert spaces generated by the bilinear mapping is considered.b-frame operator is defined; analogues of some well-known results of frame theory are obtained in Hilbert spaces. Conditions for the existence ofb-frame in Hilbert spaces are obtained; the relationship between the definite bounded surjective operator andb-frame is also studied. The concept ofb-orthonormalb-basis is introduced.

scholarly journals Controlled Continuous ∗ - K - g -Frames for Hilbert C ∗ -Modules

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Abdeslam Touri ◽  
Hatim Labrigui ◽  
Mohamed Rossafi ◽  
Samir Kabbaj
Keyword(s):  
Hilbert Spaces ◽  
Frame Theory

Frame theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert C ∗ -modules. In this paper, we define and study the new concept of controlled continuous ∗ - K - g -frames for Hilbert C ∗ -modules and we establish some properties.

scholarly journals Perturbation and Stability of Continuous Operator Frames in Hilbert C ∗ -Modules

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Abdeslam Touri ◽  
Hatim Labrigui ◽  
Mohamed Rossafi ◽  
Samir Kabbaj
Keyword(s):  
Hilbert Spaces ◽  
Continuous Operator ◽  
Frame Theory ◽  
The Stability

Frame theory has a great revolution in recent years. This theory has been extended from the Hilbert spaces to Hilbert C ∗ -modules. In this paper, we consider the stability of continuous operator frame and continuous K -operator frames in Hilbert C ∗ -modules under perturbation, and we establish some properties.

DUAL PAIRS OF GABOR FRAMES FOR TRIGONOMETRIC GENERATORS WITHOUT THE PARTITION OF UNITY PROPERTY

2011 ◽  
Vol 04 (04) ◽  
pp. 589-603 ◽  
Author(s):  
O. Christensen ◽  
Mads Sielemann Jakobsen
Keyword(s):  
Hilbert Spaces ◽  
Partition Of Unity ◽  
Frame Theory ◽  
Gabor Frames ◽  
Series Expansions ◽  
Dual Frames ◽  
Dual Pairs ◽  
Orthonormal Bases ◽  
Explicit Constructions

Frames is a strong tool to obtain series expansions in Hilbert spaces under less restrictive conditions than imposed by orthonormal bases. In order to apply frame theory it is necessary to construct a pair of so called dual frames. The goal of the article is to provide explicit constructions of dual pairs of frames having Gabor structure. Unlike the results presented in the literature we do not base the constructions on a generator satisfying the partition of unity constraint.

Bessel Sequences and Its F-Scalability

2015 ◽  
Vol 7 (4) ◽  
pp. 441-453 ◽  
Author(s):  
Lei Liu ◽  
Xianwei Zheng ◽  
Jingwen Yan ◽  
Xiaodong Niu
Keyword(s):  
Quantum Mechanics ◽  
Wavelet Analysis ◽  
Frame Theory ◽  
Riesz Bases ◽  
Gabor Analysis ◽  
Perturbation Result ◽  
Extension Principles ◽  
Bessel Sequences

AbstractFrame theory, which contains wavelet analysis and Gabor analysis, has become a powerful tool for many applications of mathematics, engineering and quantum mechanics. The study of extension principles of Bessel sequences to frames is important in frame theory. This paper studies transformations on Bessel sequences to generate frames and Riesz bases in terms of operators and scalability. Some characterizations of operators that mapping Bessel sequences to frames and Riesz bases are given. We introduce the definitions of F-scalable and P-scalable Bessel sequences. F-scalability and P-scalability of Bessel sequences are discussed in this paper, then characterizations of scalings of F-scalable or P-scalable Bessel sequences are established. Finally, a perturbation result on F-scalable Bessel sequences is derived.

scholarly journals g-Bases in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Xunxiang Guo
Keyword(s):  
Hilbert Spaces ◽  
Schauder Basis ◽  
Orthonormal Bases ◽  
Schauder Bases

The concept ofg-basis in Hilbert spaces is introduced, which generalizes Schauder basis in Hilbert spaces. Some results aboutg-bases are proved. In particular, we characterize theg-bases andg-orthonormal bases. And the dualg-bases are also discussed. We also consider the equivalent relations ofg-bases andg-orthonormal bases. And the property ofg-minimal ofg-bases is studied as well. Our results show that, in some cases,g-bases share many useful properties of Schauder bases in Hilbert spaces.

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