Mathematical Sciences: Orthogonal Polynomials: Applications and Computation

这项研究的主要目的是研究正交多项式的计算和构造方面。应特别注意生成Sobolev型正交多项式的构造方法的发展和分析，即关于涉及导数的内积的正交的多项式。它们最初被同时引入到近似函数及其导数中，此后受到了相当大的关注，但主要是从逼近理论和代数的角度出发，很少考虑计算方面的问题。其他要考虑的问题包括计算有限区间上关于Hermite权的多项式的正交性以及特殊求积规则的余项的分析。正交多项式是逼近的基本工具。它们提供对函数的正交展开和最小二乘近似，并有助于各种求积过程。此外，它们还广泛应用于常微分方程组的数值求解和解线性代数方程组的迭代方法的应用。

The main objective of this research is the study of computational and constructive aspects of orthogonal polynomials. Special attention is to be given to the development and analysis of constructive methods for generating orthogonal polynomials of Sobolev type, i.e., polynomials that are orthogonal with respect to an inner product involving derivatives. These have originally been introduced to approximate functions and their derivative simultaneously, and have since received a considerable amount of attention but largely from approximation-theoretic and algebraic points of view and with little regard to computational aspects. Other topics to be considered involve the question of computing polynomials orthogonal relative to a Hermite weight on a finite interval and the analysis of the remainder terms of special quadrature rules. Orthogonal polynomials are basic tools of approximation. They provide orthogonal expansions and least square approximations to functions and contribute to a variety of quadrature processes. In addition they are used extensively in the numerical solution of ordinary differential equations and in the application of iterative methods for solving systems of linear algebraic equations.