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

9401180巴格特这个项目的目的是继续阐述与小波基和多分辨分析这一重要领域有关的概念和定理的摘要。该项目的一部分是得到关于小波和MRA的存在性、小波和MRA的性质以及小波展开的可和性结果的结果。在抽象的背景下，人们可以运用强大的现代么正表示理论、von Neumann代数等。本项目的另一部分是使用抽象理论来描述加窗傅立叶变换的概念，该概念也可以用么正表示理论和非对易调和分析来表示。小波理论的研究是数学中一个非常活跃的领域。这主要是由于数据压缩和模式识别等涉及高性能计算的重要领域的强大应用。小波理论通常是用实变量调和分析技术来研究的。这个项目的目的是在抽象的数学背景下建立小波理论。然后，可以应用强大的抽象数学来得到关于小波的自动存在和收敛定理。对小波基本性质的新见解将会出现。以一种互惠的方式，小波的显著特征将被反映出来，以产生抽象数学中的新概念。***

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. ***