Cubature Formulae and Orthogonal Polynomials in Several Variables

摘要：该项目继续一个正在进行的计划，其目的是建立一个可行的理论，正交多项式（OP）在几个变量，并使用知识来构建容积公式（CF），同义词高维数值积分公式。P.I.制定了一个总体框架。在这个计划的第一阶段，包括一个广泛的理论OP在多个变量的基础上一个新的向量矩阵符号和系统的研究之间的关系CF和公共零点的（准）OP。这个项目的主要重点是OP和CF上的单位球，单位球和标准单形的欧几里德空间。出发点是P.I.最近的观察。这些整环上的正交结构是密切相关的，这导致了对这些经典整环上的OP的新的理解，并为构造CF提供了一种强有力的新方法。特别注意OP和CF的结构，它们在某些群下是不变的，如八面体群或单形的对称群，这与最近发展的与反射群相关的h-谐波有密切关系。立方公式和多元正交多项式与应用数学的许多分支如数值积分、逼近、编码理论、数据拟合、微分方程数值解、有限元法等有着密切的联系。CF本身对于高维积分的实际计算是必不可少的，高维积分是数值分析中的基本问题之一，并且经常被作为高速计算中的测试问题。本项目寻求在几个变量的CF和OP的性质的新的理解。其目的是确定球面、球和单形上正交结构之间的精确关系，特别是解析关系，从而使正交展开式的收敛性有新的进展，并发展出实用的方法，在这些区域上，特别是在单位球面上，产生新的有效的数值积分公式。

Abstract: The project continues an ongoing program, the aim of which is to establish a workable theory for orthogonal polynomials (OP) in several variables and use the knowledge to construct cubature formulae (CF), synonym for higher dimensional numerical integration formulae. A general framework has been developed by the P.I. in the first phase of this program, including an extensive theory of OP in several variables based on a new vector-matrix notation and a systematic study of the relation between CF and the common zeros of (quasi-)OP. The main focus of this project is on OP and CF on theunit sphere, on the unit ball and on the standard simplex of the Euclidean space. The starting point is a recent observation made by the P.I. that orthogonal structures on these domains are closely related, which has led to new understanding about OP on these classical domains and to a powerful new method for constructing CF. Special attention will be given to the structure of OP and CF that are invariant under certain groups, such as octahedral groupor symmetric group of the simplex, which has a close relation to the recent development of h-harmonics associated to the reflection groups. Cubature formulae and orthogonal polynomials in several variables havefruitful connections with many branches of applied mathematics such as numerical integration, approximation, coding theory, data fitting, numericalsolution of differential equation, finite element methods to name a few. CF itself is essential for practical evaluation of high dimensional integrals, which is one of the basic questions in numerical analysis and is often taken as a test problem in high speed computing. The present project seeks new understanding of the nature of CF and OP in several variables. Its aim is to determine the precise relationship between orthogonal structures on the sphere,the ball, and the simplex, especially analytic relations which will lead to new progress on convergence of the orthogonal expansion, and to develop practicalmethod that will yield new effective numerical integration formulae on these domains, especially on the unit sphere.