连续框架和局部紧群上的框架表示

The frame theory is an important topic in functional analysis and wavetes, which has both theoretical value and wide applications. This project will focus on the theoretical research, where the mail tools include vector measures, vector integrals (Bochner integral, Pettis integral, Dunford integral ), operator theory and some aspects for von Neumann algebras and group representations. The main research contents read as follows: (1) By using the generalized Kothe function spaces, we will investigate the theory of continuous frames in Banach spaces. The first aim is to infer necessary and /or sufcient conditions for an measurable function to be an continuous frames, and the second one is to establish the reconstruction formula for integrals. (2) By the means of direct integrals of Hilbert spaces, we introduce and study continuous operator valued frames in Hilbert spaces. The intrinsic connections between such frames and positive operator valued measures and the dilation theorem for dual pairs of such frames will be investigated. (3) Given a continuous frame, we will construct a discrete frame for which some properties (e.g. reconstruction formula, frame bounds) of original will be preserved well. (4) Combining with the left (right) regular representations, we will study the continuous frame representations for locally compact (abelian) groups.

框架理论是泛函分析和小波分析的重要研究内容，既有理论价值又有广泛应用。本项目着重于理论研究，主要工具是向量值 (算子值) 测度和积分(Bochner积分、Pettis积分、Dunford积分)、算子理论, 以及von Neumann代数和群表示的某些知识。研究内容包括：(1) 以广义Kothe函数空间为分析空间, 研究Banach空间上的连续框架，目的是建立可测函数成为连续框架的(充分或必要)条件以及积分重构公式的存在性; (2) 以 Hilbert空间的直接积分为工具，研究连续算子值框架，包括它与算子值测度的联系和对偶框架对的延展问题; (3) 给定连续框架，如何将其离散化得到离散框架并尽可能好地保持原框架的某些性质 (如重构公式, 框架界); (4) 结合左（右）正则表示，研究局部紧 (交换) 群上的连续框架表示。

框架理论是泛函分析和小波分析的重要研究内容，在信息通信等领域具有广泛应用，用算子理论与算子代数的方法研究框架是近几年的研究热点。本项目主要研究内容包括：Banach空间上的连续框架、Hilbert空间上的连续算子值框架、向量值和算子值连续框架的离散化、局部紧群上的连续框架表示、框架斜对偶的提升、算子值Gabor框架及Gabor对偶等...本课题属于基础数学理论研究。我们按计划完成了本课题的主要工作，取得若干有价值的研究成果，达到了预期目的。 通过本项目的研究，我们对框架理论与算子理论、算子代数的联系有了更深刻的理解，积累了一些有效的研究方法，对今后进一步开展相关研究具有重要的作用.