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扩展的数学理论的谐波和功能分析中的研究和教育活动。小波和框架理论在数学上不仅作为研究的主题很有趣，而且该领域在纯数学之外发现了许多应用，从应用和计算谐波分析到信号处理和数据压缩。小波是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。该项目的更广泛影响与谐波分析和小波领域的本科生和研究生的教育和培训有关。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建造大量非各向异性扩展语音的良好定位正交小波的技术。密切相关的互补话题是对无膨胀语音的小波的研究。 PI和Speegle对小波集问题的最新解决方案，描述了存在最小支持的频率（MSF）小波的语音，与数量的几何形状相连，更具体地，估计了球的扩张量的晶格点。该项目的另一个方向是建造具有所需属性的框架，例如规定的规范和框架操作员。这一研究与Schur-horn定理的无限维概括密切相关。表征自相邻操作员对角线的问题不仅对框架理论有影响，而且在von Neumann代数的环境中也进行了广泛的研究。最后，PI旨在调查Akemann-Weaver的猜想，这是Weaver猜想的更高范围扩展，由Marcus，Spielman和Srivastava证明，他们突破了Kadison-Singer问题的突破性解决方案。这一奖项通过评估了NSF的宣传，并以评估范围来表现出色的范围。