Mathematical Sciences: Special Functions and Orthogonal Expansions with Applications to Nonlinear Dynamics and Quantum Mechanics/Orthogonal Expansions and Potential Theory

这个项目的工作将集中在正交多项式理论中出现的问题上。这些是一个复变量的多项式，它们相对于复平面上定义的某个密度(度量)是正交的。将特别强调那些在有界集上受支持的度量。这种设置比研究定义在圆上的多项式更一般。它的动机是对薛定谔方程和动力学问题的研究，这些问题导致了Julia集上的多项式正交，以及关于具有无界或周期项的Jacobi矩阵的谱问题。后一个问题出现在统计物理学中。这项工作的两个主要目标涉及正交多项式的根的位置和分析涉及多项式的连续根随着次数的增加而涉及的某些渐近关系。所采用的方法涉及一种潜在的理论方法，通过支持集的格林函数来衡量渐近性。其他工作将涉及定义在平面区域上的勒贝格空间的Pade逼近和加权多项式逼近。

Work on this project will focus on problems arising in the theory of orthogonal polynomials. These are polynomials in one complex variable which are orthogonal with respect to some density (measure) defined on the complex plane. Particular emphasis will be placed on those measures which are supported on bounded sets. The setting is more general than the study of polynomials defined on a circle. It is motivated by studies of the Schrodinger equation and dynamical problems which have led to polynomials orthogonal on Julia sets and on spectral questions concerning Jacobi matrices with unbounded or periodic entries which are orthogonal on several intervals. The latter questions emerging from statistical physics. Two primary goals in the work involve the location of the roots of orthogonal polynomials and analyzing certain asymptotic relationships involving successive roots of polynomials as their degrees increase. The method to be employed involves a potential theoretic approach by which asymptotics are measured by the Green's function for the support set. Other work will involve Pade approximation and weighted polynomial approximation in Lebesgue spaces defined on plane regions.