Some Problems in Orthogonal Polynomials and Wavelets

正交多项式和小波的一些问题

• 批准号: 9970613
• 负责人: Jeffrey Geronimo
• 金额: $ 7.87万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2003-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970613&HistoricalAwards=false
• 关键词: Some; Problems; Orthogonal; Polynomials; Wavelets

Proposal: DMS-9970613Principal Investigator: Jeffrey S. GeronimoAbstract: Geronimo will study polynomials orthogonal on several subarcs of the unit circle in order to determine the analog in that context of the Toda flow associated with polynomials orthogonal on finite segments of the real line. He will continue to investigate the properties of two-dimensional pseudopolynomials orthogonal on the bi-circle and their connection to two-dimensional extension problems. This will be related to the factorization of trigonometric polynomials in two variables, a topic that has important applications in wavelet theory. Geronimo also plans to use intertwining analysis to investigate the theory and construction of orthogonal wavelet bases in one and higher dimensions. He will use the results obtained to help study orthogonal multiresolution analyses for quasicrystals.Wavelets and orthogonal polynomials have proven themselves to be important tools in attacking complex, real-world problems such as the analysis and compression of images and signals (like those found on the internet), forecasting on the basis of fragmentary data, and speech modeling. This is because these mathematical objects either have a simple structure (in the case of wavelets) and/or possess special properties (as orthogonal polynomials do) that can be exploited to decompose complex problems of the types just mentioned into simpler, more manageable ones. For instance, a family of wavelets is obtained by scaling and shifting one particular carefully chosen function, which makes it extremely useful for studying images at various resolutions. Taking advantage of orthogonality, on the other hand, is the most practical way to retrieve information that has a "prescribed direction." In order for wavelets and orthogonal polynomials to be truly useful, however, one must know how to build these functions and must understand their properties as fully as possible. The object of the basic research described above is to contribute to the achievement of both these goals.

建议：DMS-9970613首席研究员：Jeffrey S.Geronimo摘要：Geronimo将研究单位圆的几个子圆弧上的多项式正交，以便确定与实线的有限段上的多项式正交相关的Toda流的上下文中的模拟。他将继续研究双圆上正交的二维伪多项式的性质及其与二维扩张问题的联系。这将与两个变量的三角多项式的因式分解有关，这是一个在小波理论中有重要应用的主题。Geronimo还计划使用交织分析来研究一维和更高维的正交小波基的理论和构造。他将利用所获得的结果来帮助研究准晶的正交多分辨率分析。小波和正交多项式已被证明是解决复杂的现实世界问题的重要工具，例如图像和信号的分析和压缩(如在互联网上找到的那些)、基于零碎数据的预测以及语音建模。这是因为这些数学对象要么具有简单的结构(在小波的情况下)和/或具有特殊的性质(如正交多项式)，可以利用这些性质将刚才提到的类型的复杂问题分解成更简单、更易于管理的问题。例如，一族小波是通过对特定的精心选择的函数进行缩放和平移而获得的，这使得它对于研究各种分辨率的图像非常有用。另一方面，利用正交性是检索具有“指定方向”的信息的最实用方法。然而，为了让小波和正交多项式真正有用，必须知道如何构造这些函数，并且必须尽可能充分地了解它们的性质。上述基础研究的目的是为实现这两个目标作出贡献。