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。小波和框架理论本身不仅在数学上很有趣，而且在纯数学之外也有很多应用，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。该项目旨在回答小波和框架理论中一些最基本的问题。该项目的主要研究方向之一是开发用于构造大类别非各向同性膨胀的良好定域正交小波的技术。一个密切相关的补充课题是研究非膨胀膨胀的小波。最近用PI和Speegle对具有最小支持频率（MSF）小波存在的膨胀体的小波集问题的解，与数的几何有关，更具体地说，与球的膨胀体的点阵数的估计有关。项目的另一个方向是构造具有规定规范和框架算子等期望属性的框架。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具有框架理论的意义，而且在冯诺依曼代数的环境下也得到了广泛的研究。最后，PI旨在研究Akemann-Weaver猜想，这是由Marcus， Spielman和Srivastava在他们对Kadison-Singer问题的突破性解决中证明的Weaver猜想的高阶扩展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project involves research and education activities in harmonic and functional analysis concerning the mathematical theory of multi-dimensional wavelet and frame expansions. Wavelet and frame theory is not only mathematically interesting as a subject of the study by itself, but this area has found many applications outside of pure mathematics ranging 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 the education and training of undergraduate and graduate students in the area of harmonic analysis and wavelets.The project aims to answer 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 the construction of well-localized orthogonal wavelets for large classes of non-isotropic expanding dilations. A closely related complementary topic is the study of wavelets for non-expanding dilations. A recent solution of the wavelet set problem by the PI and Speegle, 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. 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 to 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 PI aims to investigate the Akemann-Weaver conjecture, which is a higher-rank extension of Weaver’s conjecture that was proven by Marcus, Spielman, and Srivastava in their breakthrough solution of the Kadison-Singer problem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.