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。这包括分析多项式和相关的量，如高阶导数、Christoffel函数、Cotes数和在度量(确定正交性)的奇点上的正交多项式的根。相关工作还将研究衡量标准中绝对连续部分消失的点，以及这些数量在支撑点附近的性质。这项工作将应用于统计分布的尾部估计。当给出正交多项式作为雅可比矩阵的特征函数时，相应的度量是该矩阵的谱度量。度量的绝对连续分量(Szego条件)的对数可积性是不可控的。这个项目的一个主要目标是找到关于矩阵的充要条件，以便满足可积性条件。常数雅可比矩阵具有纯奇异谱测度的紧扰动的例子已经被发现。关于相关反问题的工作正在进行中：找出任何谱度量的条件，以确定它是否源于常量雅可比矩阵的紧扰动。还将努力将经典的Szego理论推广到复平面上的曲线上支承的度量的情况，研究经典Lebesgue空间中(而不是逐点的)正交多项式的行为，并继续研究具有相应谱度量的多项式的三项递归定义中的相关系数。计划在离散时间理论中的最小二乘预测理论、平稳随机过程和信号处理中的应用。

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.