Hilbert 空间上框架扰动的新结果

该文给出Hilbert空间中框架扰动的新结果,也讨论Riesz 基, 近- Riesz 基和Riesz 框架的扰动问题. 所得结果包含一些已知扰动结果.     

Banach空间中的X_d框架与Reisz基

本文引入并研究了Banach空间中的X_d框架,X_d Bessel列,紧X_d框架,独立X_d框架和X_d Riesz基等概念,给出了X_d框架和独立X_d框架的算子等价刻画,Banach空间X中存在X_d框架或X_d Riesz基的充要条件以及X_d框架的对偶框架存在的充要条件,讨论了Banach空间的基和X_d框架,X_d Riesz基之间的关系．     

Hilbert空间中的g-Riesz框架

g-框架作为Hilbert空间中的推广框架最近被提出,它们有许多和框架类似的性质,但并不是所有的性质都是相似的.Christensen已指出了每个Riesz框架都包含一个Riesz基.本文指出并不是所有的g-Riesz框架都包含一个g-Riesz基,但我们得到了每个g-Riesz框架都包含一个无冗g-框架,同时给出了H...     

Banach空间上的N—框架与M—Riesz基

本文首先在Banach空间引入了N－框架与M－Riesz基。给出N－框架的充要条件和N－框架与M－Riesz基的关系,其中M,N为Orilicz函数,再讨论它们的稳定性。     

广义框架和广义Riesz基的摄动

引入并研究了Banach空间X中的Bessel集、广义框架与广义Riesz基．对X中的任一Bessel集{gm}m∈M，定义有界线性算子T：L^2（P）→X^＊，利用算子丁，给出了Bessel集与广义框架的等价刻画．同时讨论了广义框架和广义Riesz基的摄动．     

Banach空间上p-fusion框架的若干等价描述

本文说明Banach空间上p-fusion框架和p-框架有紧密联系.应用分析算子和合成算子给出p-fusion Bessel序列、p-fusion框架和q-fusion Riesz基的等价描述.     

Hilbert空间中的K-Riesz框架及其稳定性

K-框架是Hilbert空间框架的一种推广.本文融合K-框架和Riesz框架的思想,提出了KRiesz框架和K-Riesz基的概念,分别得到了K-Riesz基和K-Riesz框架的等价刻画,并给出K-Riesz框架异于Riesz框架的一个性质.最后,借助框架理论的方法和技巧,给出了Hilbert空间中K-Riesz框架和K-Riesz基的若干个稳定性结论.     

Banach空间上的框架与Riesz基

本文讨论Banach空间上框架、无冗框架与Riesz基之间的关系及它们的稳定性．     

一类乘子算子在全测度集上的逼近

本文给出了研究乘子算子在全测度集上逼近的一种框架.在 Riesz 极大算子有界的条件下,确定了一类乘子算子在 Riesz 位势空间上几乎处处逼近的阶.并用于讨论广义 Bochner-Riesz 平均和 Abel-Cartwright 平均的点态逼近.     

复Hilbert空间中的K-框架和K-Riesz基

复Hilbert空间中的K-框架是框架的一种推广,是Gǎvruta在研究算子K的原子分解系统时引入的.本文首先在Hilbert空间H中引入K-Riesz基的概念,给出H中K-Riesz基界为A和B的K-Riesz基的两个等价刻画及K-框架界为A和B的K-框架的一个特征.众所周知,H中无冗框架与Riesz基是等价的,但是无冗K-框架与K-Riesz基是不等价的.接着研究H中无冗K-框架与K-Riesz基之间的关系.最后,考虑H中K-框架或K-Riesz基的扰动的稳定性.当K为H中的恒等算子时,这些结果与框架或Riesz基的相应结果是一致的.     

Fusion-Riesz frame in Hilbert space

Fusion-Riesz frame (Riesz frame of subspace) whose all subsets are fusion frame sequences with the same bounds is a special fusion frame. It is also considered a generalization of Riesz frame since it shares some important properties of Riesz frame. In this paper, we show a part of these properties of fusion-Riesz frame and the new results about the stabilities of fusion-Riesz frames under operator perturbation (simple named operator perturbation of fusion-Riesz frames). Moreover, we also compare the operator perturbation of fusion-Riesz frame with that of fusion frame, fusion-Riesz basis (also called Riesz decomposition or Riesz fusion basis) and exact fusion frame.     

Banach空间上的框架与拟Riesz基

该文首先给出Ｂａｎａｃｈ空间上的框架与拟Ｒｉｅｓｚ基的充要条件,其次讨论Ｂａｎａｃｈ空间上的框架和拟Ｒｉｅｓｚ基的稳定性,特别地,讨论在Ｂａｎａｃｈ空间上关于框架与拟Ｒｉｅｓｚ基的广义Ｐａｌｅｙ Ｗｉｅｎｅｒ定理．     

Operator Frames for Banach Spaces

In this paper, operator Bessel sequences, operator frames, Banach operator frames, Operator Riesz bases for Banach spaces and dual frames of an operator frame are introduced and discussed. The necessary and sufficient condition for a Banach space to have an operator frame, a Banach operator frame or an operator Riesz basis are given. In addition, operator frames and operator Riesz bases are characterized by the analysis operator of operator Bessel sequences.     

On the Riesz fusion bases in Hilbert spaces

In this paper we investigate the connection between fusion frames and obtain a relation between indexes of the synthesis operators of a Besselian fusion frame and associated frame to it. Next we introduce a new notion of a Riesz fusion bases in a Hilbert space. We show that any Riesz fusion basis is equivalent with a orthonormal fusion basis. We also obtain generalizations of Theorem 4.6 of [1]. Our results generalize results obtained for Riesz bases in Hilbert spaces. Finally we obtain some results about stability of fusion frame sequences under small perturbations.     

Angle criteria for frame sequences and frames containing a Riesz basis

We obtain a condition implying that the union of two frame sequences is also a frame sequence. Christensen found a condition for this in terms of orthogonal projections. We phrase our condition by use of the angle between closed subspaces. Also a lower bound formula is obtained. We then apply the results to find conditions for a frame containing a Riesz basis to be a Riesz frame.     

On Various R-duals and the Duality Principle

The duality principle states that a Gabor system is a frame if and only if the corresponding adjoint Gabor system is a Riesz sequence. In general Hilbert spaces and without the assumption of any particular structure, Casazza, Kutyniok and Lammers have introduced the so-called R-duals that also lead to a characterization of frames in terms of associated Riesz sequences; however, it is still an open question whether this abstract theory is a generalization of the duality principle. In this paper we prove that a modified version of the R-duals leads to a generalization of the duality principle that keeps all the attractive properties of the R-duals. In order to provide extra insight into the relations between a given sequence and its R-duals, we characterize all the types of R-duals that are available in the literature for the special case where the underlying sequence is a Riesz basis.     

Frames and the Feichtinger conjecture

We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linearly independent sequences. We further show that the two conjectures are implied by the paving conjecture. Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences.