Mathematical Sciences: Orthogonal Polynomials

这个项目的工作将集中在与正交多项式相关的数学分析主题上。这项工作有两个方向。首先，我们可以从实直线或单位圆上的一个度量开始，然后考虑相应的正交多项式及其递归关系。相反，人们可以从由递归关系产生的一类多项式开始，并询问关于它们变得与之正交的度量的信息。在后一种情况下，产生递归系数的度量的存在已经知道一段时间了。这里要做的工作是以一种计算高效的方式从系数中确定度量。另一项研究将研究与度量相关的Christoffel函数的渐近行为。克里斯托弗函数与统计预测理论有关。它们使与正交多项式族相关的多项式积分最小化。此外，还将分析用于研究中心力场中薛定谔方程的无穷大三对角矩阵的特征值。具体目标包括改进单位圆上与Szego类度量相关的Christoffel函数极限的一个不等式，并缩小正交多项式理论与信号处理之间的差距。预计结果将在电子信号的处理和传输中发挥越来越大的作用。

Work on this project will focus on topics in mathematical analysis connected with orthogonal polynomials. There are two directions the work takes. First one can begin with a measure on the real line or unit circle and consider the corresponding orthogonal polynomials and their recurrence relations. Conversely, one may begin with a class of polynomials arising from a recurrence relation and ask for information about measures against which they become orthogonal. In this latter context, the existence of measures yielding the recurrence coefficients has been known for some time. Work to be done here is that of determining the measure from the coefficients in a computationally efficient way. Another line of investigation will pursue the asymptotic behavior of the Christoffel function associated with a measure. Christoffel functions are related to statistical prediction theory. They minimize polynomial integrals associated with families of orthogonal polynomials. In addition, work will be done analyzing eigenvalues of infinite tri-diagonal matrices used in the study of Schrodinger equations in central force fields. Specific goals include improving an inequality for the limits of Christoffel functions on the unit circle associated with measures of the Szego class and narrowing the gap between the theory of orthogonal polynomials and signal processing. Results are expected to play an increasing role in the processing and transmission of electronic signals.