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）、多元求积公式（CFs） 和 它们在多元近似中的应用。 它包含三个完整的部分。第一部分涉及调查的定性和定量特征的OPEs的权重函数不变的某些群体下的各种规则域，如立方体，球，单纯形，和领域。 目标是揭示底层域的几何形状如何影响OPEs的属性， 探讨 多元正交多项式的渐近性质，并建立了多元正交多项式的各种乘子定理。 第二部分的研究涉及多元CF，这将是研究的帮助下，多元正交多项式。该项目呼吁新的多元CF的建设方法，如通过研究多个变量的多项式插值。最后一部分是多元逼近，这将与OPEs和CFs的应用一起研究。所得结果将直接应用于插值、数值分析、特殊函数和偏微分方程。本文的主要研究工具是谐波分析、泛函分析和Dunkl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.