Cubature Formulae and Orthogonal Polynomials of Several Variables

DMS奖摘要奖#：0201669PI：Xu，袁研究所：俄勒冈大学项目：应用数学项目经理：Catherine Mavriplis标题：立方公式和多元正交多项式本项目的目的是研究立方公式(CF)、高维数值积分公式的同义词和多元正交多项式(OP)的各个方面。该项目要求从代数的角度研究体积公式，以及新的构造方法，如通过研究多元多项式插值。它还建议进一步研究OP关于各种正则区域上的权函数的性质，如立方体、球、单纯形和球体。多元立方体公式(CF)和正交多项式(OP)与数值积分、逼近、编码理论、数据拟合、组合学、统计学等多个应用数学分支，以及调和分析、群表示、计算代数几何、微分方程式等领域都有着丰富的联系，其目的是进一步揭示CF和OP的隐含性质，寻找封闭公式和显式构造，从而建立具有实际意义的理论结果和方法。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。该项目试图从几个变量中寻求对循环现金流量和运筹股性质的新理解，并旨在获得具有实际意义的可行结果，例如构建循环现金流量的新方法。日期：2002年6月18日

DMS Award AbstractAward #: 0201669PI: Xu, YuanInstitution: University of OregonProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Cubature Formulae and Orthogonal Polynomials in Several VariablesThe aim of this project is to study various aspects of cubature formulae (CF), synonym for higher dimensional numerical integration formulae, and orthogonal polynomials of several variables (OP). The project calls for a study of cubature formula from an algebraic point of view as well as new constrction method, such as through studying of polynomial interpolation in several variables. It also proposes to investigate further properties of OP with respect to weight functions on various regular domains, such as the cube, the ball, the simplex and the sphere. The aim is to uncover further hidden properties of CF and OP, and to find closed formulae and explicit construction, so that further theoretic results and methods with practical implication can be built upon.Cubature Formulae (CF) and Orthogonal Polynomials (OP) of several variables have fruitful connections with many branches of applied mathematics such as numerical integration, approximation, coding theory, data fitting, combinatorics and statistics, to name a few, as well as with areas such as harmonic analysis, group representation, computational algebraic geometry and differential equation. 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 proposed project seeks new understanding in the nature of CF and OP in several variables, and it aims at workable results that have practical implications, such as new method for constructing CF. Date: June 18, 2002