Mathematical Sciences: Orthogonal Polynomials

本专题将讨论与正交多项式相关的数学分析的各种主题。在同一区间上，有许多类的正交多项式。在每一类中，正交性是根据集中在区间上的度量（密度）来定义的。这项工作的主要主题是继续努力理解测度和相应多项式的递归系数之间的关系。与每一类正交多项式相关联的是一个独特的算法，它用低阶多项式产生高阶多项式。这个递归式是一个简单的线性关系，关系中的系数是非常有趣的。所采用的方法包括最小化多项式积分（Christoffel函数）、差分方程的渐近性和无限三对角矩阵的特征值。用于研究矩阵特征值的方法可以适用于研究薛定谔方程在中心力场中的解，Christoffel函数与统计预测理论有关。这项工作的自然副产品多项式近似的结果和预测理论在电子信号的数字处理中有价值。

Various topics in mathematical analysis associated with orthogonal polynomials will be taken up in this project. On the same interval, there are many classes of orthogonal polynomials. In each class, orthogonality is defined with respect to a measure (density) concentrated on the interval. The main theme of this work is a continuing effort to understand the relationship between the measure and the recurrence coefficients of the corresponding polynomials. Associated to each class of orthogonal polynomials is a unique algorithm which produces the higher order polynomials in terms of the lower. This recurrence is a simple linear relation, and it is the coefficients in the relation that are of considerable interest. Methods to be employed include minimizing polynomial integrals (the Christoffel function), asymptotic behavior of difference equations and eigenvalues of infinite tridiagonal matrices. The methods used for studying eigenvalues of matrices can be adapted to the study of solutions of the Schrodinger equation in a central field of force, and the Christoffel function is related to statistical prediction theory. Results of polynomial approximation, a natural by-product of this work, and prediction theory are valuable in digital processing of electronic signals.