Mathematical Sciences: Orthogonal Polynomials: Applications and Computation

本研究的主要目标是研究正交多项式的计算和建设方面。 特别注意的是要给予发展和分析的建设性方法产生正交多项式的索伯列夫型，即，关于包含导数的内积正交的多项式。 这些最初被引入到近似函数及其衍生物的同时，并已收到了相当多的关注，但主要是从近似理论和代数的观点，很少考虑到计算方面。 其他要考虑的议题涉及的问题，计算多项式正交相对于埃尔米特重量在有限的时间间隔和分析的剩余条款的特殊正交规则。 正交多项式是逼近的基本工具。 它们提供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，它们被广泛用于常微分方程的数值解和线性代数方程组的迭代求解。

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.