Operator Algebras and Wavelet Theory

摘要：这项建议的计划是利用算符-代数方法来解决小波数学理论中的几个问题，以及框架理论中的相关领域，包括Weyl-Heisenberg理论和Gabor理论。我们在众所周知的小波连通性问题上有了新的线索，我们认为这是本课题的一个基本问题。另一个基本问题性质的小波问题涉及已知为MRA小波的线性组合的Riesz小波何时本身就是MRA小波的问题。我们建议研究的其他问题包括小波框架的范数密度，小波、超框架和超小波的算子理论内插，以及Weyl-Heisenberg框架。我们还将致力于两个不同方向的项目，涉及算子空间、自反性和最优化。第一个涉及抽象算子空间的排序函数的公理描述。对于具体给定的算子空间，有一个具体排序函数的自然定义，但对此的抽象描述是难以捉摸的，似乎是一个基本问题。第二个问题是关于在最优化理论意义上哪些矩阵完成问题是适定的。这项建议代表了关于小波和框架理论数学的跨学科性质的工作。以前得到NSF支持的这一方向的工作已经解决了一些悬而未决的问题，并影响了其他人在调和分析和面向应用的小波理论方面的工作。继续朝这一方向前进是本提案的主旨。在过去的十几年里，已经写了许多关于小波在信号和图像处理中的应用的论文。到目前为止，大多数已发表的工作都是直接与应用程序有关的，而关于该学科的基本数学基础方面的成果相对较少。这项建议与这一数学有关。在过去的两年里，我们的工作中出现了一些令人惊讶的事情，这些发现导致了一些以前没有被怀疑的潜在应用领域。我们在这项提案中强调了几个突出问题，并计划开展工作。在之前的NSF支持下，我们还与几位合著者一起完成了关于算子理论、算子空间、算子代数和矩阵优化的一些基本问题的研究，这导致了一些进一步的开放问题和我们计划追求的方向。有几个研究生和本科生参与了这个项目。这笔赠款还包括美国国家科学基金会对一年一度的大平原算符理论研讨会的部分支持，这是一个在美国多所大学之间轮流举行的大型数学研究会议。

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.