Multivariate Approximation

用更简单的函数表示或逼近复杂的数据，允许用各种方法分析和处理数据。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，即依赖于多个变量的函数。为了使多变量技术有效，通常需要与单变量函数的相应方法有很大的不同。本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。短期目标包括边界连续可微的多变量域上代数多项式近似误差的表征，多变量凸域上Christoffel函数行为的几何表征，多变量域一般类别上的正培养公式。

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.