Mathematical Sciences: Orthogonal Polynomials

这个项目的工作将继续研究定义在实数和复数的子集上的正交多项式的性质。正交性是关于某种度量(密度)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合。多项式可以在不同的情况下出现--作为微分方程解或作为递推关系的结果。但这一措施始终存在于幕后。重点将放在理解与几个不相交区间上所支持的度量正交的多项式。当度量被支持在单个区间上时，多项式的渐近公式以积分表示的形式得到。我们将努力将这项工作推广到几个区间上的支集的情况。另一条研究方向将考虑多项式关于复平面上Julia集上支承的测度的正交性。提出了两个目标。首先证明了当Julia集是Cantor集时，含有这些多项式的Jacobi矩阵一般是有限周期的。第二个目标是通过递归公式确定正交性度量的矩。一种新的研究方法是利用遍历理论研究单位圆上的正交多项式的性质。当递推关系由随机过程产生时，存在Lyaponov指数。当指数为零时，有证据表明该度量有绝对连续的部分。将努力核实这一点。

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.