Harmonic and Functional Analysis of Wavelet and Frame Expansions

小波和框架展开的谐波和泛函分析

基本信息

批准号：
1956395
负责人：
Marcin Bownik
金额：
$ 21.1万
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：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1956395&HistoricalAwards=false
关键词：
Harmonic Functional Analysis Wavelet Frame

项目摘要

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.

该项目涉及调和和泛函分析方面的研究和教育活动，涉及多维小波和框架展开的数学理论。小波和框架理论不仅作为一个研究课题本身在数学上很有趣，该领域还发现了纯数学之外的许多应用，从应用和计算调和分析到信号处理和数据压缩。小波作为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。该项目更广泛的影响包括调和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开发用于大类非各向同性膨胀扩张的良好定域正交小波的构造技术。一个密切相关的补充主题是非扩张扩张的小波研究。存在最小支持频率（MSF）小波的膨胀特征问题与数的几何有关；更具体地说，是对球膨胀的格点数量的估计。存在良好局域化小波的膨胀分类问题对非各向同性函数空间的研究具有重要意义。该项目的另一个方向是建造具有某些所需属性的框架，例如规定的规范和框架操作员。这一研究方向与 Schur-Horn 定理的无限维推广密切相关。表征自伴算子对角线的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成果。该奖项反映了 NSF 的法定使命，并通过利用基金会的智力价值和更广泛的影响进行评估，认为值得支持审查标准。