Cubature Formulae and Orthogonal Polynomials in Several Variables

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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.