Harmonic and functional analysis of wavelet and frame expansions

小波和框架展开的调和和泛函分析

• 批准号: 2349756
• 负责人: Marcin Bownik
• 金额: $ 24.6万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349756&HistoricalAwards=false
• 关键词: Harmonic; functional; analysis; wavelet; frame

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.

该项目涉及有关多维小波和框架展开的数学理论的调和和泛函分析的研究和教育活动，小波和框架理论不仅作为研究主题本身在数学上很有趣，而且该领域已经找到了许多应用。纯数学之外，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例子包括 JPEG 2000 数字图像标准和用于数据存储的指纹压缩。随着教育和培训调和分析和小波领域的本科生和研究生。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构建技术的开发。大类非各向同性膨胀扩张的良好局域化正交小波的研究 一个密切相关的补充主题是 PI 和 Speegle 最近对小波集问题的研究。存在最小支持频率（MSF）小波的膨胀与数字几何有关，更具体地说，与球膨胀的格点数量的估计有关。该项目的另一个方向是构建具有所需的框架。这一研究方向与 Schur-Horn 定理的无限维推广密切相关。自伴算子对角线的特征问题不仅对框架理论有影响。但它也在冯·诺依曼代数的背景下得到了广泛的研究。最后，PI 旨在研究阿克曼-韦弗猜想，这是韦弗猜想的更高阶扩展，已由马库斯、斯皮尔曼和斯里瓦斯塔瓦在他们的论文中证明。 Kadison-Singer 问题的突破性解决方案。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。