Orthogonal expansions, cubature formulas and approximation in several variables

正交展开、体积公式和多变量近似

• 批准号: 311678-2010
• 负责人: Dai, Feng
• 金额: $ 1.75万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=508770
• 关键词: Orthogonal; expansions; cubature; formulas; approximation

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.

该计划建议解决涉及多元正交多项式扩展（OPES），多元立方体公式（CFS）及其在多元近似中的应用的问题。 它包含三个集成部分。第一部分涉及对OPE的定性和定量特征的调查，这些特征在某些常规域（例如立方体，球，单纯形和球体）上的某些组上的重量功能不变。 目标是揭示基础域的几何形状如何影响OPES的性质，研究多元正交多项式的渐近性质，并为OPES建立变化的乘数定理。 研究的第二部分涉及多元CFS，将在多元正交多项式的帮助下进行研究。该项目呼吁采用新的多元CFS构建方法，例如通过研究多个变量的多项式插值研究。研究的最后一部分是多元近似，将与OPES和CFS的应用一起研究。结果将直接应用于插值，数值分析，特殊功能和部分微分方程。拟议研究的主要工具是来自谐波分析，功能分析和H-Harmonics的Dunkl理论。