Orthogonal expansions, cubature formulas and approximation in several variables

本课程主要研究多元正交多项式展开式（OPEs）、多元文化公式（CFs）及其在多元逼近中的应用。它包含三个完整的部分。第一部分研究了OPEs在各种正则域（如立方体、球、单纯形和球）上的权函数不变群下的定性和定量特征。目的是揭示基础域的几何如何影响OPEs的性质，研究多元正交多项式的渐近性质，并建立OPEs的各种乘数定理。研究的第二部分是多元CFs，它将借助多元正交多项式进行研究。本课题需要新的多元CFs的构造方法，例如通过研究多个变量的多项式插值。研究的最后一部分是多元逼近，它将与OPEs和CFs的应用一起研究。其结果将直接应用于插值、数值分析、特殊函数和偏微分方程。本文研究的主要工具是谐波分析、泛函分析和邓克尔h-谐波理论。

This program proposes to work on problems involving multivariate orthogonal polynomial expansions (OPEs), multivariate cubature formulas (CFs) and their applications to multivariate approximation. It contains three integrated parts. The first part concerns investigations into qualitative and quantitative features of OPEs with respect to weight functions invariant under certain groups on various regular domains, such as the cube, the ball, the simplex, and the sphere. The goals are to reveal how geometry of the underlying domain influences the properties of OPEs, to investigate asymptotic properties of multivariate orthogonal polynomials, and to establish variaous multiplier theorems for OPEs. The second part of the research concerns multivariate CFs, which will be studied with the help of multivariate orthogonal polynomials. This project calls for new construction methods of multivariate CFs, such as through the study of polynomial interpolation in several variables. The last part of the research is multivariate approximation, which will be studied together with applications of OPEs and CFs. The results will have direct applications to interpolation, numerical analysis, special functions and partial differential equations. The main tools for the proposed research are from harmonic analysis, functional analysis and the Dunkl theory of h-harmonics.