Harmonic and Functional Analysis of Wavelet and Frame Expansions

该项目涉及谐波和泛函分析的研究和教育活动，涉及多维小波和框架展开的数学理论。小波和框架理论不仅在数学上很有趣，而且在纯数学之外也有很多应用。这些范围从应用和计算谐波分析到信号处理和数据压缩。小波作为关键工具的一些著名示例包括JPEG 2000数字图像标准和用于数据存储的指纹压缩。该项目更广泛的影响涉及谐波分析和小波领域的本科生和研究生的教育和培训。这个活动有三个组成部分，它们与这个项目的研究目标紧密相连。一是对本项目研究方向的博士生进行指导和培养。第二个教育目标是支持学生组织的随机/泛函分析研讨会，这将为对谐波和泛函分析研究感兴趣的研究生提供额外的、关键的培训。第三个目标是招募高级本科生参与框架研究项目。该项目旨在回答小波和框架理论中一些最基本的问题。本项目的主要研究方向之一是开发用于大类别非各向同性膨胀膨胀的良好定域正交小波的构造技术。一个密切相关的补充课题是研究非膨胀膨胀的小波。存在最小支持频率（MSF）小波的膨胀特性问题与数的几何性质有关；更具体地说，用对球的膨胀的晶格点的数目的估计。一类具有良好定域小波的膨胀分类问题在非各向同性函数空间的研究中具有重要意义。项目的另一个方向是构造具有期望属性的框架，如具有规定的规范和框架算子。这条研究路线与舒尔-霍恩定理的无限维推广密切相关。自伴随算子对角线的刻画问题不仅对框架理论有重要意义，而且在冯诺依曼代数的环境中也得到了广泛的研究。最后，描述紧密融合框架序列的问题与代数组合方法有关，而代数组合方法通常不用于分析。

The project involves research and education activities in harmonic and functional analysis concerning the mathematical theory of multidimensional wavelet and frame expansions. Wavelet and frame theory is not only mathematically interesting in its own right, but this area has found many applications outside of pure mathematics. These range from applied and computational harmonic analysis to signal processing and data compression. Some well known examples where wavelets are a key tool include the JPEG 2000 digital image standard and fingerprint compression for data storage. The broader impacts of the project deal with education and training of undergraduate and graduate students in the area of harmonic analysis and wavelets. There are three components of this activity which are integrally connected with the research aims of this project. The first is advising and training of Ph.D. students in the research directions described in this project. The second educational goal is a support for student-run Stochastic/Functional Analysis seminar, which will provide additional, critical training for graduate students interested in doing research in harmonic and functional analysis. The third aim is to recruit advanced undergraduate students to research projects on frames. The project aims at answering some of the most fundamental questions in wavelet and frame theory. One of the main research directions of the project is the development of techniques for construction of well-localized orthogonal wavelets for large classes of nonisotropic expanding dilations. A closely related complementary topic is the study of wavelets for nonexpanding dilations. The problem of characterizing dilations for which there exist minimally supported frequency (MSF) wavelets is connected with the geometry of numbers; more specifically, with the estimate on the number of lattice points of dilates of balls. An analogous problem of classifying dilations for which there exist well-localized wavelets has implications in the study of nonisotropic function spaces. Another direction of the project is the construction of frames with desired properties, such as with prescribed norms and frame operator. This line of research is closely related with the infinite-dimensional generalizations of the Schur-Horn theorem. The problem of characterizing diagonals of self-adjoint operators has not only implications for frame theory, but it has also been extensively studied in the setting of von Neumann algebras. Finally, the problem of characterizing tight fusion frame sequences is connected with algebraic combinatorics methods that are typically not employed in analysis.