Multivariate Approximation

多元近似

• 批准号: RGPIN-2020-05357
• 负责人: Prymak, Andriy
• 金额: $ 1.31万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758565
• 关键词: Multivariate; Approximation

Representation or approximation of complicated data by simpler functions permits various methods for analysis and processing of the data. Real life information usually depends on many parameters and thus can naturally be described by multivariate functions, i.e. those depending on several variables. For a multivariate technique to be effective, one usually needs a significant departure from the corresponding approaches for functions of one variable. The main long-term objective of the proposed research program is to continue study of multivariate approximation with emphasis on approximation by algebraic polynomials on multivariate domains, properties of orthogonal polynomials and discretization results. Short-term objectives include characterization of the error of approximation by algebraic polynomials on multivariate domains with continuously differentiable boundary, geometric characterization of the behaviour of Christoffel function on multivariate convex domains, positive cubature formulas on general classes of multivariate domains. The research has primarily a theoretical flavor, and it is anticipated that the results will be applicable in approximation theory, harmonic analysis, functional analysis, convex geometry and other areas of mathematics and possibly other related disciplines. Some outcomes will also be useful for data science, numerical analysis and computer aided geometric design. The highly qualified personnel trained will gain skills of conducting independent mathematical research and obtain knowledge in the related areas mentioned above.

用更简单的函数表示或近似复杂的数据允许采用各种方法来分析和处理数据。现实生活中的信息通常取决于许多参数，因此可以自然地用多元函数（即取决于多个变量的函数）来描述。为了使多变量技术有效，通常需要与一个变量函数的相应方法有很大的不同。拟议研究计划的主要长期目标是继续研究多元近似，重点是多元域上的代数多项式近似、正交多项式的性质和离散化结果。短期目标包括在具有连续可微分边界的多元域上用代数多项式来表征近似误差、在多元凸域上对 Christoffel 函数的行为进行几何表征、在一般类型的多元域上使用正立方公式。该研究主要具有理论风味，预计研究结果将适用于逼近论、调和分析、泛函分析、凸几何和其他数学领域以及可能的其他相关学科。一些成果对于数据科学、数值分析和计算机辅助几何设计也很有用。培养的高素质人才将获得进行独立数学研究的技能并获得上述相关领域的知识。