Multivariate Approximation

用更简单的函数表示或近似复杂的数据允许使用各种方法来分析和处理数据。现实生活中的信息通常依赖于许多参数，因此自然可以用多元函数来描述，即那些依赖于几个变量的函数。为了使多变量技术有效，通常需要与相应的单变量函数方法有很大的不同。提出的研究计划的主要长期目标是继续研究多元逼近，重点是多元区域上代数多项式的逼近，正交多项式的性质和离散化结果。短期目标包括刻画具有连续可微边界的多元区域上代数多项式逼近的误差、多元凸域上Christoffel函数的性质的几何刻画、一般多元区域上的正体积公式。 这项研究主要具有理论色彩，预计其结果将适用于逼近理论、调和分析、泛函分析、凸几何和其他数学领域，并可能适用于其他相关学科。一些成果也将对数据科学、数值分析和计算机辅助几何设计有用。 培训的高素质人员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。

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.