Mathematical Sciences: Orthogonal Polynomials: Applications and Computation

本研究的主要目的是研究正交多项式的计算和构造方面。特别注意的是发展和分析生成Sobolev型正交多项式的构造方法，即多项式与涉及导数的内积正交。它们最初被引入近似函数及其导数，并得到了相当多的关注，但主要来自近似理论和代数的观点，很少考虑计算方面。其他要考虑的问题包括计算有限区间上相对于Hermite权值的正交多项式的问题，以及分析特殊正交规则的余项。正交多项式是逼近的基本工具。它们提供了函数的正交展开式和最小二乘近似，并有助于各种正交过程。此外，它们还广泛用于常微分方程的数值解和求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.