Operator Algebras and Wavelet Theory

算子代数和小波理论

• 批准号: 0070796
• 负责人: David Larson
• 金额: $ 14.11万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070796&HistoricalAwards=false
• 关键词: Operator; Algebras; Wavelet; Theory

ABSTRACT:The plan of this proposal is to utilize operator-algebraic methods to solve several problems in the mathematical theory of wavelets, and in the related area of frame theory, including Weyl-Heisenberg and Gabor theory. We have a new lead on the well-known wavelet connectivity problem, which we perceive to be a basic issue in the subject. Another wavelet problem of a basic issue nature concerns the question of when a Riesz wavelet which is known to be a linear combination of MRA wavelets is itself an MRA wavelet. Others problems we propose to work on concern norm-density of the wavelet frames, operator-theoretic interpolation of wavelets, superframes and superwavelets, and Weyl-Heisenberg frames. We will also work on two projects in a different direction concerning operator spaces, reflexivity and optimization. The first concerns an axiomatic description of a ranking-function for an abstract operator space. There is a natural definition of a concrete ranking-function for a concretely given operator space, but an abstract description of this is elusive and seems to be a basic issue. The second concerns the question of which matrix completion problems are well-posed in the sense of optimization theory. This proposal represents work of an interdisciplinary nature on the mathematics of wavelet and frame theory. Work in this direction that was previously supported by NSF has settled some open questions and has impacted the work of others in harmonic analysis and applications-oriented wavelet theory. Continuing in this direction is the main thrust of the present proposal. Numerous papers have been written in the past dozen years dealing with applications of wavelets to signal and image processing. So far, most of the published work has dealt directly with applications, and relatively little has been accomplished concerning the basic mathematical underpinnings of the subject. This proposal is concerned with this mathematics. There have been some surprises that have come up in our work in the past two years, and these discoveries have led to some potential areas of applications of a previously unsuspected nature. There are several outstanding problems we emphasize in this proposal and plan to work on. Under prior NSF support we also accomplished research with several co-authors on some basic problems in operator theory, operator spaces, operator algebras and matrix optimization, and this leads to some further open problems and directions we plan to pursue. Several graduate and undergraduate students arinvolved in this project. This grant also contains the NSF partial support of the annual Great Plains Operator Theory Symposium, a major mathematics research conference which rotates among a number of universities in the USA.

摘要：该计划的计划是利用算子代数方法来解决小波数学理论中的几个问题，以及框架理论的相关领域，包括Weyl-Heisenberg和Gabor理论。我们有一个新的领导著名的小波连通性问题，我们认为这是一个基本问题的主题。 另一个基本问题性质的小波问题涉及的问题时，Riesz小波，这是已知的MRA小波的线性组合本身是一个MRA小波。 其他问题，我们建议工作的关注范数密度的小波框架，小波，超框架和超小波，和Weyl-Heisenberg框架的算子理论插值。 我们还将致力于两个项目在一个不同的方向，涉及算子空间，自反性和优化。 第一个问题涉及一个抽象算子空间的秩函数的公理化描述。 对于一个具体的算子空间，有一个具体的秩函数的自然定义，但对它的抽象描述是难以捉摸的，似乎是一个基本问题。 第二个问题是矩阵完备问题在最优化理论的意义上是适定性的。 这项建议代表了工作的一个跨学科性质的数学小波和框架理论。以前由NSF支持的这个方向的工作已经解决了一些悬而未决的问题，并影响了其他人在谐波分析和面向应用的小波理论方面的工作。继续朝这个方向努力是本建议的主要方向。 在过去的十几年中，已经有大量的论文涉及小波在信号和图像处理中的应用。到目前为止，大多数已发表的工作直接处理的应用程序，并已完成相对较少的基本数学基础的问题。 这个建议与数学有关。 在过去的两年里，我们的工作中出现了一些令人惊讶的发现，这些发现导致了一些以前未知的潜在应用领域。 有几个突出的问题，我们强调在这个建议，并计划工作on. Under以前NSF的支持下，我们还完成了研究与几个合作者在算子理论，算子空间，算子代数和矩阵优化的一些基本问题，这导致了一些进一步的开放问题和方向，我们计划追求。 几个研究生和本科生参与了这个项目.该补助金还包含NSF对年度大平原算子理论研讨会的部分支持，该研讨会是一个主要的数学研究会议，在美国的一些大学中轮流举行。