Mathematical Sciences: Orthogonal Polynomials

这个项目的工作将继续研究在实数和复数子集上定义的正交多项式的性质。正交性是根据一些度量（密度）来定义的，根据该度量的性质，人们会遇到完全不同的多项式集。多项式可能出现在不同的环境中——作为微分方程的解或作为递归关系的结果。但这一措施总是存在于背景之中。特别的重点将放在理解多项式正交的措施支持在几个不相交的区间。多项式的渐近公式已经发展为积分表示，当测度被支持在一个单一的区间。将作出努力，每隔一段时间把这项工作延续到支助的情况。另一条研究路线将考虑多项式与复平面上Julia集合上支持的测度正交。提出了两个目标。首先是证明当Julia集合是康托集合时雅可比矩阵的多项式是有限周期的。第二个目标是通过递归公式确定正交测度的矩。利用遍历理论研究单位圆上正交多项式的性质是一种新的研究方法。当递归关系由随机过程产生时，存在李亚波诺夫指数。当指数为零时，有证据表明该度量具有绝对连续的部分。将作出努力核实这一点。

Work on this project will continue investigations into the nature of orthogonal polynomials defined on subsets of the real and complex numbers. The orthogonality is defined with regard to some measure (density) and, depending on the nature of that measure, one encounters radically different sets of polynomials. The polynomials may arise in different contexts - as solutions of differential equations or as the result of recurrence relations. But the measure always exists in the background. Particular emphasis will be placed on understanding polynomials orthogonal to measures supported on several disjoint intervals. Asymptotic formulas for polynomials have been developed in terms of integral representations when measures are supported on a single interval. Efforts will be made to carry over this work to the case of support on several intervals.Another line of investigation will consider polynomials orthogonal with respect to measures supported on Julia sets in the complex plane. Two goals are set forth. The first is to show that in general the Jacobi matrix with these polynomials is limitperiodic when the Julia set is a Cantor set. A second goal is to determine the moments of the orthogonality measure via the recurrence formulas. A new line of investigation uses ergodic theory to study properties of orthogonal polynomials on the unit circle. When the recurrence relation is generated by a stochastic process, there is a Lyaponov exponent. When the exponent is zero, there is evidence that the measure has an absolutely continuous part. Efforts will be made to verify this.