Harmonic and functional analysis of wavelet and frame expansions

该项目涉及谐波和泛函分析方面的研究和教育活动，涉及多维小波和框架扩展的数学理论。小波和框架理论不仅是数学上有趣的研究本身的主题，但这一领域已经发现了许多应用范围以外的纯数学应用和计算谐波分析的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。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非各向同性膨胀膨胀构造良好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非扩展膨胀的小波研究。最近的解决方案的小波集问题的PI和Speegle，特征膨胀，存在最小支持频率（MSF）小波，是连接到几何的数字，更具体地说，与估计的数目的格点的球的膨胀。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。刻画自伴算子的对角线问题不仅对框架理论有影响，而且在冯诺依曼代数的背景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Kadison-Singer问题的突破性解决方案中证明的Weaver猜想的更高级别扩展。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

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.