Wavelets, Frames and Group Representations

摘要在过去的二十年里，小波理论已经发展成为数学和应用科学的一个基础领域。密切相关的是框架理论和采样理论，它们都与谐波分析有关。提出的研究将研究群体表示作为一个统一的主题，因为它们出现在小波，框架和抽样理论的设置。韦伯的研究将使用泛函和调和分析的现有工具，同时也开发与群环和极大伴代数相关的工具作为这些对象的模型。拟议的研究本质上是理论性的，尽管其中大部分是由应用数学中的问题推动的。韦伯的研究目的是为小波和框架理论建立一个坚实的数学基础。然而，这项研究将潜在地影响应用数学和信号处理方面的问题，如数字信号复用、冗余通信和安全通信。

Proposal Number: DMS-0200756PI: Eric WeberABSTRACTIn the past twenty years, wavelet theory has developedinto a fundamental area of mathematics and applied science.Intimately related are frame theory and sampling theory, allwith a connection to harmonic analysis. The proposedresearch will investigate group representations as a unifying theme since they appear in the settings of wavelet,frame, and sampling theory. The research of Weber will useexisting tools from functional and harmonic analysis, whilealso developing tools related to group cocycles and maximalabelian self adjoint algebras as models of these objects.The proposed research is theoretical in nature, though muchof it is motivated by problems in applied mathematics. Thegoal of the research of Weber is to establish a solidmathematical foundation of wavelet and frame theory. However,the research will potentially impact problems in appliedmathematics and signal processing, such as digital signal multiplexing, redundant communications and securecommunications.