Cubature Formulae and Orthogonal Polynomials in Several Variables

摘要：该项目是一项正在进行的计划的继续，其目的是建立一种可行的多变量正交多项式(OP)理论，并利用这些知识来构造立方公式(CF)，它是高维数值积分公式的同义词。在这个项目的第一阶段，P.I.已经制定了一个通用的框架，包括基于新的向量矩阵记法的多变量OP的广泛理论，以及系统地研究CF与(准)OP的公共零点之间的关系。这个项目的主要焦点是单位球面上的OP和CF，单位球上的单位球和欧几里德空间的标准单形。本文的出发点是P.I.最近的一项观察，即这些区域上的正交结构是密切相关的，这导致了对这些经典区域上的OP的新的理解，并为构造CF提供了一种强有力的新方法。我们将特别关注在某些群下不变的OP和CF的结构，例如八面体群或单形的对称群，这与反射群相关的h-调和的最近发展有密切的关系。立方公式和多元正交多项式与数值积分、逼近、编码理论、数据拟合、微分方程数值求解、有限元方法等应用数学的许多分支有着密切的联系。高维积分是数值分析中的一个基本问题，也是高速计算中的一道试题。本项目试图从几个变量中寻求对CF和OP的性质的新理解。它的目的是确定球面、球面和单纯形上的正交结构之间的精确关系，特别是将导致正交展开式收敛的新的解析关系，并发展实用的方法，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.