Cubature Formulae and Orthogonal Polynomials of Several Variables

DMS Award AbstractAward #： 0201669 PI： 徐元研究所： 俄勒冈大学课程： 应用数学项目经理：Catherine Mavriplis职务：本项目的目的是研究体积公式（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容积公式以及新的插值方法，例如通过研究多元多项式插值。 它还提出了进一步的研究性质的OP关于各种规则域，如立方体，球，单纯形和球体上的权函数。多元立方公式（CF）和多元正交多项式（OP）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、数值积分等有着密切的联系，它们的研究和应用也是一个重要的领域，因此，本文对多元立方公式和多元正交多项式的研究和应用进行了深入的探讨。组合学和统计学，仅举几例，以及调和分析，群表示，计算代数几何和微分方程等领域。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。该项目旨在对CF和OP在几个变量中的性质进行新的理解，并旨在获得具有实际意义的可行结果，例如构建CF的新方法。日期：二○ ○二年六月十八日

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