Mathematical Sciences: Some Problems in Harmonic Analysis Related to Wavelets

数学科学：与小波相关的调和分析中的一些问题

• 批准号: 9204323
• 负责人: Michael Frazier
• 金额: $ 4万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204323&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Problems; Harmonic

The thrust of this mathematical research revolves around applications of recent developments in wavelet theory. The first application will focus on algebras of linear operators generated by those of Calderon-Zygmund type. These operators arise in the analysis of partial differential equations. The operators are too complex to study individually; the goal of current research is to choose wavelet bases in which the operators may be represented as almost-diagonal matrices. From this representation, one can read off many hidden properties of the operators. A second line of investigation concerns wavelet decompositions in higher dimensional space. This work will begin with the representation of radial functions in term of wavelets which somehow respect the symmetry. The fundamental difficulty is that one cannot symmetrize one-dimensional wavelets and maintain the desired translation properties. Some progress has been made in developing a new wavelet-type function with many desirable properties preserved. A third direction of this research is that of distinguishing the behavior of the different Riesz transforms on functions spaces using wavelet-type representations. Wavelet theory has introduced an extraordinary new tool into the field of harmonic analysis. It contains the elements of a highly effective theoretical method for the representation of functions along with natural algorithms for the computation of various expansions of functions which are simultaneously local in time and frequency. There applications are only beginning to be felt in the scientific and engineering community.

这项数学研究的主旨是 小波理论最新发展的应用。 第一 应用程序将集中在代数的线性算子生成 卡尔德隆-齐格蒙德类型的人。 这些运算符出现在 偏微分方程分析 经营者 太复杂而无法单独研究;当前研究的目标 是选择小波基，其中算子可以是 表示为几乎对角矩阵。 从这个 表示，人们可以读出许多隐藏的属性， 运营商 第二条调查路线涉及小波 在高维空间中的分解。 这项工作将开始 用小波表示径向函数 它以某种方式尊重对称性。 最根本的困难 一维小波不能对称化， 保持所需的翻译属性。 了一些进展 在开发一种新的小波型函数， 保存的理想属性。 第三个方向 研究是区分不同的行为， 函数空间上的小波型Riesz变换 表示。 小波理论引入了一个非凡的新工具， 谐波分析领域。 它包含一个 一种高效的理论方法， 函数沿着自然算法， 同时在局部的函数的各种扩展， 时间和频率。 这些应用才刚刚开始 在科学和工程界的感受。