时频分析与算子代数

Gabor分析中几个著名的基本定理(如对偶原理和稠密性定理)与群表示和算子代数理论密切相连.尽管时频分析与算子代数之间的某些联系是Jon von Neumann于1930年代建立的,可是它们在近期得到广泛研究,这主要应归于小波/Gabor理论或更一般的框架理论近二十年的发展.本文将讨论过去几年得到的一些主要结果,同时也给出一些新的结果、解释和问题,我们主要考虑来源于时频分析并能反映与群表示理论存在内在联系的那些结果.特别地,针对群表示的时频分析,将详细说明抽象的对偶原理及其与算子代数理论中几个公开问题的联系.     

紧支撑Gabor框架的稳定性

1引言自Duffin和Schaeffer提出了框架理论以来,框架理论在数学和信息科学等领域占有非常重要的地位,在小波分析和时频分析中起着举足轻重的作用.现在Gabor框架系统被广泛地应用在信息论,量子力学,信号处理和图象处理等方面.Gabor框架的稳定性是应用中所需要的,同时也是人们关心的问题.但是,一直以来对Gabor框架的稳定性的研究主要集中在L~2(R)上,即使在L~2(R~d)上的研究也主要针对单项指标扰动进行了研究.本文借助于文[2]中的定理3和定理4,文[3]中的定理2和文[5]中的定理2.1,分别对L~2(R~d)上紧支撑Gabor框架的窗函数、平移指标、旋转指标以及多项混合扰动的稳定性进行了讨论.     

由多重贝塞尔序列扩充为仿射框架

仿射框架在信号处理中有实用性.运用算子理论与时频分析,将两个二重贝塞尔序列扩充为一对对偶二重仿射框架.再由已知的一对多重贝塞尔序列添加若干个函数使它们扩充为一对对偶多重仿射框架,得到了多重Gabor框架的特征不等式.     

关于rectifiable空间中的局部(序列)连通性的几个注记

给出连通的rectifiable空间是局部序列连通(或局部连通)的刻画,推广了拓扑群中的相应结果;利用rectifiable空间G中e的局部邻域基给出G是局部连通(或局部序列连通)的刻画;证明了若A是rectifiable空间G中的序列开子集,那么H=A是G的序列开rectifiable子空间.     

连通和序列紧的Rectifiable空间（英文）

本文主要讨论了rectifiable空间的连通,序列紧和κ-Frechet-Urysohn性质.证明了以下结果:(1)若G是局部σ-序列紧且具有Souslin性质的rectifiable空间,则G是σ-序列紧的.(2)每一连通的局部σ-紧的rectifiable空间G是σ-紧的.(3)若rectifiable空间G的每一紧(可数紧,序列紧)的子空间是Frechet-Urysohn,则G的每一紧(可数紧,序列紧)的子空间是强Frechet-Urysohn.这些结果推广了拓扑群中的相应结果.     

多个生成子生成的Gabor框架

本文首先给出了由多个生成子生成的Gabor系成为Gabor框架(多Gabor框架)的几个充分条件.然后利用上述充分条件之一,构造出了一类多Gabor框架,它的生成子具有很好的时-频局部性质,此构造结果规避了Balian-Low定理的限制.另外,文中又给出了两种构造方案,通过这些方案可以对已有的多Gabor框架进行修正从而得到许多新的多Gabor框架.     

L2(Rd)的Gabor框架的扰动

本文研究了L2(Rd)上以矩阵平移和调制的Gabor框架的扰动,得到了若干有意义的结果.     

Banach空间中同时逼近问题的适定性

研究一般Banach空间X中同时逼近问题的适定性.对严格凸的KadecBanach空间X中的相对有界弱紧闭子集G,建立了关于最佳同时逼近问题适定Bair纲结果.进一步,当X是一致凸空间时,证明了E(G)中使其最佳同时逼近问题不适定的序列在E(G)中是一个σ-多孔集.另外,还研究了关于最佳同时逼近元具有分歧域的集合G的几乎性.     

Hilbert空间中的g-Bessel序列的一些等式与不等式(II)

Hilbert 空间中的g- 框架是框架的自然推广, 它们包含了许多推广的框架, 如子空间框架或fusion 框架、斜框架和拟框架等. 它们有许多与框架类似的性质, 但是并不是所有的性质都是相似的.例如, 无冗框架等价于Riesz 基, 但是无冗g- 框架不等价于g-Riesz 基. 一些作者将Hilbert 空间中的框架和对偶框架的等式和不等式推广到g- 框架和对偶g- 框架. 本文建立Hilbert 空间中的g-Bessel序列或g- 框架的一些新的等式和不等式. 本文还给出这些不等式的等号成立的充要条件. 这些结果推广和改进了由Balan, Casazza 和G?vruta 等得到的著名结果.     

半直线伸缩调制框架集

本文研究右半直线平方可积函数空间L^2(R+)中的一类伸缩调制系.实际问题中时间变量不可取负值,L^2(R+)可模拟因果信号空间.但因R+按加法不能作成一个群,它不容许小波与Gabor系.我们研究L^2(R+)中由特征函数生成的伸缩调制系(MD-系)框架,引入了R+中MD-框架集的概念,利用"伸缩等价"与"基数函数"方法刻画了L^2(R+)中MD-Bessel集与完备集;得到了关于MD-Riesz基集的两个充分条件,并证明了通过对MD-Riesz基集进行有限可测分解可得到MD-框架集.     

Duality Principles in Frame Theory

Duality principles in Gabor theory such as the Ron–Shen duality principle and the Wexler–Raz biorthogonality relations play a fundamental role for analyzing Gabor systems. In this article we present a general approach to derive duality principles in abstract frame theory. For each sequence in a separable Hilbert space we define a corresponding sequence dependent only on two orthonormal bases. Then we characterize exactly properties of the first sequence in terms of the associated one, which yields duality relations for the abstract frame setting. In the last part we apply our results to Gabor systems.     

Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to display the deeper properties of Gabor frames. Furthermore, we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) to show that the Morita-Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the existence of good multi-window Gabor frames for modulation spaces with Gabor atoms in e.g. Feichtinger's algebra or in Schwartz space.     

Sufficient conditions for irregular Gabor frames

Finding general and verifiable conditions which imply that Gabor systems are (resp. cannot be) Gabor frames is among the core problems in Gabor analysis. In their paper on atomic decompositions for coorbit spaces [H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations, and their atomic decomposition, I, J. Funct. Anal. 86 (1989), 307–340], the authors proved that every Gabor system generated with a relatively uniformly discrete and sufficiently dense time-frequency sequence will allow series expansions for a large class of Banach spaces if the window function is nice enough. In particular, such a Gabor system is a frame for the Hilbert space of square integrable functions. However, their proof is based on abstract analysis and does not give direct information on how to determine the density in the sense of directly applicable estimates. It is the goal of this paper to present a constructive version of the proof and to provide quantitative results. Specifically, we give a criterion for the general case and explicit density for some cases. We also study the existence of Gabor frames and show that there is some smooth window function such that the corresponding Gabor system is incomplete for arbitrary time-frequency lattices.     

Banach空间的渐近赋范性质与（S）性质

本文证明渐近赋范性质与（ｓ）性质具有对偶性，并且给出一个Ｂａｎａｃｈ空间为ＨａｈｎＢａｎａｃｈ光滑的充要条件。     

Very Convex Banach Spaces

ＶｅｒｙＣｏｎｖｅｘＢａｎａｃｈＳｐａｃｅｓＴｅｇｕｓｉ（特古斯）Ｓｕｙａｌａｔｕ（苏雅拉图）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＩｎｎｅｒＭｏｎｇｏｌｉａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｈｕｈｈｏｔ，０１００２２）ＬｉＹｏｎｇｊｉｎ...     

Banach空间中线性流形上的度量投影的表达式

借助于正规对偶映射,建立了一般Banach空间中线性流形上的(集值)度量投影存在的 充要条件,同时给出了度量投影的表达式和点到线性流形上的距离公式.这些本质地推广和改进了 王玉文和于金凤在空间自反、严格凸和光滑强假定下的相应结果.     

Norming Subspaces Isomorphic to l_1

Norming subspaces are studied widely in the duality theory of Banach spaces. These subspaces are applied to the Borel and Baire classifications of the inverse operators. The main result of this article asserts that the dual of a Banach space X contains a norming subspace isomorphic to 1 provided that the following two conditions are satisfied:(1) X*contains a subspace isomorphic to 1;and(2) X*contains a separable norming subspace.     

Aspects of local spectral theory for positive operators

In this paper we will discuss the local spectral behaviour of a closed, densely defined, linear operator on a Banach space. In particular, we are interested in closed, positive, linear operators, defined on an order dense ideal of a Banach lattice. Moreover, for positive, bounded, linear operators we will treat interpolation properties by means of duality.Dedicated to G. Maltese on the occasion of his 60^th birthday     

次可分与拟可分的Banach空间

Ｘ是Ｂａｎａｃｈ空间，对Ｘ的每一弱可分且弱紧的子集Ｋ，若（ｋ，ω）可度量，则称Ｘ是次可分的。如果对任意的可分，则称Ｘ是拟可分的。本文证明了这两类新引入的空间比可分空间与弱紧生成的Ｂａｎａｃｈ空间都要广泛并且保留了可分空间的许多重要性质。     

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.