Multivariate Approximation

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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.