K-框架和紧K-框架的算子扰动的稳定性

该文在Hilbert空间中讨论K-框架和紧K-框架在算子扰动中的稳定性.首先给出K-框架经过有界线性算子T扰动后为K-框架的充要条件,其次讨论了用两个Bessel序列或者两个K-框架构造新的K-框架的方法,最后给出用两个Bessel序列构造紧K-框架的充要条件.这些结果推广和改进了由Christensen和Casazza等得到的著名结果.     

Hilbert空间中的K-框架

K-框架是框架理论的一种推广.K-框架可以用于重构Hilbert空间中有界线性算子值域内的元素.本文首先研究了K-框架与框架理论的关系,得到了紧K-框架成为框架当且仅当有界线性算子K是满的,给出了有界线性算子K具有闭值域的K-框架的一个充要条件.并利用有界线性算子K和合成算子构造K-框架,讨论在一定扰动条件下K-框架的稳定性.     

Hilbert空间中的广义K-框架

通过引入广义K-框架和广义K-原子系统给出了Hilbert空间上有界线性算子K的值域的一种新的重构方式.广义K-框架是Hilbert空间中广义框架和K-框架概念的一种新的推广.为了建立基于广义K-框架的元素重构理论,广义K-对偶对和广义逼近K-对偶对的概念被引入.此外,广义K-框架和广义K-原子系统之间的关系,广义K-对偶对的存在条件以及广义K-对偶对和广义逼近K-对偶对之间的联系等问题被深入地讨论和研究.     

框架的强分离性与紧框架的构造

Hilbert空间上框架的分离性概念是由Han与Larson首先提出的.分离框架有很多有用的性质.本文利用强分离紧框架来构造出新的紧框架.特别地,我们将给出一些充分或充要条件使得正规紧框架的算子系数的线性组合是某些特殊的紧框架.并给出一个框架存在界满足一定条件的紧对偶框架的充要条件.最后还讨论了实Hilbert空间中的一些相关问题.     

闭子模中K-框架的几个新的不等式

本文研究Hilbert C*-模中K-框架的不等式问题.借助K-对偶构建了闲子模中K-框架的几个新的不等式,所得结果推广和改进了Hilbert空间中框架和Hilbert C*-模中广义框架的相应结果.     

局部域上Gabor紧框架的特征

主要讨论局部域上的Gabor紧框架.首先,建立局部域上Gabor系{xm(bx)g(x-u(n)a)}m.n∈p构成L~2(K)上紧框架的特征.其次,给出Gabor系{X_m(bx)g(x-u(n)a)}_(m,n∈p)成为L~2(K)上标准正交基的充要条件.     

N维2带小波紧框架的构造

给出了$m$个函数生成$N$维2带小波紧框架的充分条件和$N$维2带小波紧框架的显式构造算法, 讨论了小波紧框架的分解算法与重构算法. 提出的构造方法很有普遍性, 容易推广到$N(N\geq2)$维$M(M\geq 2)$带小波紧框架的情形,也可以得到类似的小波紧框架的分解算法与重构算法.     

二元3带小波紧框架的构造

研究二元3带小波紧框架的结构.首先给出二元3带小波紧框架的充分条件.并给出这种小波紧框架的显式公式.若给定的尺度函数的符号函数是有理函数,则可以构造出符号函数为有理函数的小波紧框架.文中给出了数值例子,还给出了二元3带小波紧框架的分解和重构算法.     

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.     

小波紧框架的显式构造

该文研究对应于3带尺度函数的小波紧框架,这个小波紧框架是由V_1中的l个函数ψ^1, ψ^2, ψ^n 构成.给出这l个函数构成小波紧框架的充分条件.由此给出由3 带尺度函数构造出一个小波紧框架的显式公式.特别的,如果给定尺度函数的符号是有理函数,则可以构造出符号为有理函数的小波紧框架.最后还给出类似于小波的小波紧框架的分解与重构算法.      

K-fusion Frames and the Corresponding Generators for Unitary Systems

Motivated by K-frames and fusion frames, we study K-fusion frames in Hilbert spaces.By the means of operator K, frame operators and quotient operators, several necessary and sufficient conditions for a sequence of closed subspaces and weights to be a K-fusion frame are obtained, and operators preserving K-fusion frames are discussed. In particular, we are interested in the K-fusion frames with the structure of unitary systems. Given a unitary system which has a complete wandering subspace, we give a necessary and sufficient condition for a closed subspace to be a K-fusion frame generator.     

Robustness and Iterative Reconstruction of $g$-Fusion Frames in Hilbert Spaces

Regarding the application of the fusion frames and generalization of them in data proceeding, their iterative is of particular importance when one of their members is deleted. In this note, a method for reconstruction of generalized fusion frames and error operator with its upper bound are presented. Also, the approximation operator for these frames will be introduced and we study some results about them.     

Symmetric approximation of frames and bases in Hilbert spaces

We introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces. We prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames. Even in the case of the symmetric orthogonalization of bases, our techniques and results are new. A crucial role is played by whether or not a certain operator related to the initial frame or basis is Hilbert-Schmidt.

     

Generation of finite tight frames by Householder transformations

Finite tight frames are widely used for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such a purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting of m vectors in ?ⁿ or ?ⁿ only O(nm) operations are needed. In addition, we also study the following question: Given a set of vectors in ?ⁿ or ?ⁿ, how many additional vectors, possibly with constraints, does one need to add in order to obtain a tight frame?     

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.     

基于Spectral Tetris算法的框架的最佳稀疏性

当信号维数较大时,使用稀疏框架分解信号就能减少大量的加法和乘法运算,所以,研究稀疏框架很有意义.本文介绍有限框架的稀疏性,并研究基于Spectral Tetris算法构造的框架的稀疏性.首先,给出基于Spectral Tetris算法的框架的最佳稀疏性;其次,得到基于Spectral Tetris算法的可剖分紧框架的最佳稀疏性.