Orthogonal expansions, cubature formulas and approximation in several variables

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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.