Mathematical Sciences: An Abstract Theory of Wavelets and Multi-Resolution Analyses

数学科学：小波和多分辨率分析的抽象理论

• 批准号: 9401180
• 负责人: Lawrence Baggett
• 金额: $ 5.17万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-01 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401180&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Abstract; Theory; Wavelets

9401180 Baggett The purpose of this project is to continue formulating an abstract setting the notions and theorems involved with the important area of wavelet bases and multi-resolution analyses. Part of the project is to derive results about the existence of wavelets and MRA's, quality of the wavelets and MRA's, and summability results for wavelet expansions. In the abstract context, one may bring to bear the powerful and modern theories of unitary representations, von Neumann algebras, etc. Another part of the project is to use abstract theory to describe the notion of windowed Fourier transforms, which notion can also be formulated in terms of unitary representation theory and non-commutative harmonic analysis. Research on Wavelet Theory is an incredibly active area of mathematics. This is due mainly to the powerful applications to important areas involving high performance computing such as data compression and pattern recognition. Usually wavelet theory is approached using the real variable techniques of harmonic analysis. This project is aimed at formulating Wavelet Theory in an abstract mathematical setting. Powerful abstract mathematics can then be applied to get automatic existence and convergence theorems about wavelets. New insights into the fundamental nature of wavelets will occur. In a reciprocal manner, the salient features of wavelets will be reflected to produce new concepts in abstract mathematics. ***

小行星9401180 本计画的目的是继续将小波基与多解析度分析这一重要领域所涉及的概念与定理抽象化。 该项目的一部分是得出结果的小波和MRA的存在，质量的小波和MRA的，和小波扩展的可求和结果。在抽象的背景下，人们可以承担强大的和现代的理论酉表示，冯诺依曼代数等项目的另一部分是使用抽象理论来描述窗口傅立叶变换的概念，这一概念也可以制定在酉表示理论和非交换谐波分析。 小波理论的研究是一个非常活跃的数学领域。这主要是由于强大的应用程序，涉及高性能计算的重要领域，如数据压缩和模式识别。通常小波理论是用调和分析的真实的变量技术来处理的。这个项目的目的是制定小波理论在一个抽象的数学设置。强大的抽象数学，然后可以应用到自动存在性和收敛性定理小波。新的见解小波的基本性质将发生。在一个互惠的方式，小波的显着特点将反映在抽象数学产生新的概念。 ***