Mathematical Sciences: Orthogonal Polynomials and Their Applications

工作将主要集中在真实的， 复数和矩阵值正交多项式。 这包括 多项式和相关量的分析，例如 高阶导数，Christoffel函数，Cotes数 正交多项式在奇点的根 测量的支持（其确定正交性）。 相关工作还将研究绝对 连续分量的措施消失和性质 这些数量接近支持的端点。 的应用 这项工作将被用于跟踪统计数据的估计， 分布。 当正交多项式被给定为 Jacobi矩阵的特征函数，相应的测度为 矩阵的谱测度 对数 的绝对连续分量的可积性 措施（Szego的条件）是不可控的。 的一个主要目标 本项目是为了找到必要和充分条件， 矩阵，从而满足可积性条件。 常Jacobi矩阵紧扰动的例子 已经发现了具有纯奇异谱测度的。 相关的反向问题的工作正在取得进展： 条件的任何光谱测量，看看它是否来自一个 常Jacobi矩阵的紧扰动 努力将 在扩展经典Szego理论时， 在复平面中曲线上支持的测量的情况下， 研究正交多项式的行为在经典 Lebesgue空间（而不是逐点），并继续研究 三项递归定义中的相关系数 多项式与相应的谱测度。 最小二乘预测理论在离散- 时间理论平稳随机过程与信号处理 都计划好了

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.