Harmonic and Functional Analysis of Wavelet and Frame Expansions

该项目涉及关于多维小波和框架展开的数学理论的谐波和泛函分析方面的研究和教育活动。小波和框架理论不仅本身在数学上很有趣，而且在纯数学之外也有许多应用。这些范围从应用和计算谐波分析到信号处理和数据压缩。小波是关键工具的一些众所周知的例子包括JPEG 2000数字图像标准和用于数据存储的指纹压缩。该项目的更广泛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。这项活动有三个组成部分，与本项目的研究目标密切相关。一是博士生的培养与咨询;学生在这个项目中描述的研究方向。第二个教育目标是支持学生运行的随机/函数分析研讨会，这将为有兴趣在谐波和函数分析研究的研究生提供额外的，关键的培训。第三个目标是招收高年级本科生进行框架研究项目。 该项目旨在回答小波和框架理论中的一些最基本的问题。该项目的主要研究方向之一是为大类各向异性膨胀膨胀构造良好的局部化正交小波的技术的发展。一个密切相关的补充主题是研究小波的nonexpanding膨胀。存在最小支持频率（MSF）小波的特征扩张的问题与数字的几何有关;更具体地说，与球的扩张的格点的数目的估计有关。一个类似的问题，分类膨胀，存在良好的局部化小波的研究中的非各向同性函数空间的影响。该项目的另一个方向是建造具有所需特性的框架，例如具有规定的规范和框架操作员。这条研究路线与Schur-Horn定理的无限维推广密切相关。自伴算子的对角线的刻画问题不仅涉及框架理论，而且在冯诺依曼代数的背景下也得到了广泛的研究。最后，紧融合帧序列的特征化问题与通常不用于分析的代数组合学方法有关。

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.