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

[401180] Baggett这个项目的目的是继续对小波基和多分辨率分析这一重要领域所涉及的概念和定理进行抽象的表述。该项目的一部分是推导小波和MRA的存在性，小波和MRA的质量，以及小波展开的可和性结果。在抽象的背景下，人们可以承担强大的现代理论的单位表示，冯·诺伊曼代数等。项目的另一部分是使用抽象理论来描述窗口傅里叶变换的概念，这个概念也可以用单位表示理论和非交换谐波分析来表述。小波理论的研究是一个非常活跃的数学领域。这主要是由于在涉及高性能计算的重要领域（如数据压缩和模式识别）的强大应用程序。通常用谐波分析的实变量技术来研究小波理论。这个项目的目的是在抽象的数学设置中形成小波理论。然后可以应用强大的抽象数学来得到小波的自动存在性和收敛性定理。对小波基本性质的新见解将会出现。以一种相互作用的方式，小波的显著特征将被反映出来，在抽象数学中产生新的概念。＊＊＊

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. ***