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

这一项目的工作将集中于 正交多项式理论 这些都是多项式在一个 复变量是正交的关于一些 密度（测量）定义在复平面上。 特别 重点将放在那些得到支持的措施上， 有界集合 背景比研究更普遍， 定义在圆上的多项式。 它的动机是研究 薛定谔方程和动力学问题 关于Julia集和谱问题的正交多项式 关于无界或周期Jacobi矩阵 它们在几个间隔上正交。 后一个问题 从统计物理学中产生的。 工作中的两个主要目标涉及到 正交多项式的根及某些渐近分析 关系涉及多项式的连续根作为其 度增加。 所采用的方法涉及潜在的 理论方法，通过该方法， 支持集的绿色函数。 其他工作将涉及帕德近似和加权 平面上Lebesgue空间的多项式逼近 地区

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.