Cubature Formulae and Orthogonal Polynomials of Several Variables

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

DMS Award Abstract Award #: 0201669 PI: Xu, Yuan Institution: University of Oregon Program: Applied Mathematics Program Manager: Catherine Mavriplis Title: Cubature Formulae and Orthogonal Polynomials in Several Variables The 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