Harmonic and Functional Analysis of Wavelet and Frame Expansions

该项目涉及与多维小波和框架扩展的数学理论有关的谐波和泛函分析的研究和教育活动。小波和框架理论不仅是数学上有趣的研究本身的主题，这一领域也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。该项目的更广泛的影响包括教育和培训本科生和研究生在谐波分析和小波领域。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。存在最小支持频率（MSF）小波的特征膨胀的问题是连接到几何的数字，更具体地说，对球的膨胀的格点的数量的估计。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构建具有某些所需属性的框架，例如规定的规范和框架操作符。这条线的研究是密切相关的无限维推广的舒尔-霍恩定理。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，该项目涉及研究Kadison-Singer问题的突破性解决方案，旨在改进其技术并获得新的结果。该奖项反映了NSF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而被认为值得支持。

This project involves research and education activities in harmonic and functional analysis concerned with the mathematical theory of multi-dimensional wavelet and frame expansions. Wavelet and frame theory is not only mathematically interesting as a subject of study by itself, this area has also 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 this project include education and training of undergraduate and graduate students in the area of harmonic analysis and wavelets.This 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. The problem of characterizing dilations for which there exist minimally supported frequency (MSF) wavelets is connected to the geometry of numbers; more specifically, to 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 non-isotropic function spaces. Another direction of the project is the construction of frames with certain desired properties, such as prescribed norms and frame operator. This line of research is closely related to infinite dimensional generalizations of the Schur-Horn theorem. The problem of characterizing diagonals of self-adjoint operators not only has implications for frame theory, it has also been studied extensively in the setting of von Neumann algebras. Finally, the project involves investigating the breakthrough solution of the Kadison-Singer problem with the aim of improving its techniques and obtaining new results.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.