Perturbation theory for almost periodic Jacobi matrices with applications to orthogonal polynomials

几乎周期雅可比矩阵的微扰理论及其在正交多项式中的应用

• 批准号: 0901710
• 负责人: Maxim Zinchenko
• 金额: $ 9.63万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2009-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901710&HistoricalAwards=false
• 关键词: Perturbation; theory; almost; periodic; Jacobi

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The proposed project concerns perturbation theory of almost periodic Jacobi and CMV matrices with finite or infinite gap spectrum as well as asymptotic analysis of the associated orthogonal polynomials on the real line and the unit circle, respectively. The goal is to understand the relations between three fundamental objects: spectral measures (or measures of orthogonality), recursion coefficients, and orthogonal polynomials. The project investigates the interplay between regularity properties of the measures, asymptotic behavior of the coefficients, and asymptotics of the orthogonal polynomials. In particular, it extends what is known for the case of measures supported by a single interval to the case of measures supported by a union of several intervals.Problems raised in this project bring together several areas of mathematics, most notably spectral theory and the theory of orthogonal polynomials. There are also important connections to harmonic analysis and the theory of Fuchsian groups. The proposed research will enhance our understanding of periodic and almost periodic structures with and without impurities. Potential areas of application span from random matrix theory to inverse problems in computer tomography and material science. The theory of orthogonal polynomials on the unit circle has also applications in geophysical scattering and electronic circuit filter design.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本课题主要研究具有有限或无限间隙谱的概周期Jacobi矩阵和CMV矩阵的摄动理论，以及相关正交多项式在实线上和单位圆上的渐近分析。目标是理解三个基本对象之间的关系：谱测度（或正交测度）、递归系数和正交多项式。该项目研究了测度的正则性、系数的渐近性和正交多项式的渐近性之间的相互作用。特别是，它将已知的由单个区间支持的测度扩展到由多个区间的并集支持的测度。在这个项目中提出的问题汇集了几个数学领域，最著名的是谱理论和正交多项式理论。谐波分析和富克斯群理论也有重要的联系。所提出的研究将增强我们对含杂质和不含杂质的周期和几乎周期结构的理解。潜在的应用领域从随机矩阵理论到计算机断层扫描和材料科学中的逆问题。单位圆上的正交多项式理论在地球物理散射和电子电路滤波器设计中也有应用。