Some Problems in Orthogonal Polynomials and Wavelets

摘要：Geronimo将研究单位圆的几个子弧上的多项式正交，以确定与实线有限段上的多项式正交相关的Toda流上下文中的模拟。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展问题的联系。这将涉及到两个变量的三角多项式的因式分解，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高维正交小波基的理论和构造。他将利用获得的结果来帮助研究准晶体的正交多分辨率分析。小波和正交多项式已经被证明是解决复杂的、现实世界问题的重要工具，比如分析和压缩图像和信号（就像在互联网上找到的那些），基于碎片数据的预测，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），要么具有特殊的性质（如正交多项式），可以利用这些性质将刚才提到的复杂问题分解成更简单、更易于管理的问题。例如，通过缩放和移动一个特定的精心选择的函数来获得一组小波，这使得它在研究各种分辨率的图像时非常有用。另一方面，利用正交性是检索具有“规定方向”的信息的最实用的方法。然而，为了使小波和正交多项式真正有用，人们必须知道如何构建这些函数，并且必须尽可能全面地了解它们的性质。上述基础研究的目的是为实现这两个目标做出贡献。

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.