Mathematical Sciences: Orthogonal Polynomials and Their Applications

工作将主要集中在实数、复数和矩阵值正交多项式中出现的问题。这包括对多项式和相关量的分析，如高阶导数、克里斯托费尔函数、柯特数和在支持度量的奇点处正交多项式的根（这决定了正交性）。相关的工作还将着眼于测量的绝对连续分量消失的点，以及这些量在支持端点附近的性质。这项工作的应用将用于统计分布的尾部估计。当正交多项式作为雅可比矩阵的特征函数给出时，对应的测度是矩阵的谱测度。测度的绝对连续分量（Szego条件）的对数可积性是不可控的。本课题的主要目标是找到满足矩阵可积性条件的充要条件。发现了具有纯奇异谱测度的常雅可比矩阵的紧摄动的例子。有关反问题的工作正在进行中：找到任何谱测度的条件，看看它是否来自一个常数雅可比矩阵的紧摄动。还将努力扩展经典的Szego理论，使其适用于复平面上曲线支持的测度。研究正交多项式在经典勒贝格空间（而非点向）中的行为，并继续研究具有相应谱测度的多项式的三项递归定义中的相关系数。规划了离散时间理论、最小二乘预测理论、平稳随机过程和信号处理的应用。

Work will concentrate primarily on problems arising in real, complex and matrix valued orthogonal polynomials. This includes an analysis of polynomials and associated quantities such as higher order derivatives, Christoffel functions, Cotes numbers and roots of orthogonal polynomials at singular points of the support of the measure (which determines the orthogonality). Related work will also look at points where the absolutely continuous component of the measure vanishes and the nature of these quantities near the endpoints of support. Applications of this work will be made to tail estimates of statistical distributions. When orthogonal polynomials are given as eigenfunctions of Jacobi matrices, the corresponding measure is the spectral measure of the matrix. The logarithmic integrability of the absolutely continuous component of the measure (Szego's condition) is not controllable. A major goal of this project is to find necessary and sufficient conditions on the matrix so that the integrability condition will be satisfied. Examples of compact perturbations of constant Jacobi matrices having purely singular spectral measures have been discovered. Work is progressing on the relevant inverse question: to find conditions on any spectral measure to see if it arises from a compact perturbation of a constant Jacobi matrix. Efforts will also be made in extending the classical Szego theory to take in the case of measures supported on curves in the complex plane, to study the behavior of orthogonal polynomials in classical Lebesgue spaces (rather than pointwise) and to continue research on relating coefficients in three-term recursive definitions of polynomials with the corresponding spectral measures. Applications to least square prediction theory of discrete- time theory stationary stochastic processes and signal processing are planned.