Cubature Formulas, Orthogonal Expansions and Quantitative Approximation on Regular Domains

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It is often desirable to approximate a general, possibly complicated function by simpler, easier to compute functions, such as algebraic polynomials, multivariate splines and wavelets. Quantitative approximation  (QA) attempts to determine as precisely as possible the size of the error in this approximation. Cubature formulas (CFs) and orthogonal polynomial expansions(OPEs)  have been playing crucial roles in QA and many other related areas, such as numerical integration, computer tomography, coding theory, data fitting, and compressive sensing. CF itself is essential for practical evaluation of  high dimensional integrals, and  OPEs have been a main  tool for studying and constructing  CFs.  Modern problems in CFs,  OPEs and QA are often formulated in several variables on  various regular domains, such as  spheres, simplexes,  balls and cubes, driven by applications in engineering, finance, biology, medicine and quantum chemistry. ***     Many very important contributions have been made recently in  various directions of  CFs, OPEs and QA on regular domains.  Such contributions  include  the solution of the  longstanding  Korevaar-Meyer conjecture  on the optimal size of spherical designs, new world record in counterexamples to Borsuk's conjecture, universally optimal distribution of points on spheres,  characterizations of the rate of polynomial approximation in terms of  smoothness on balls, spheres, and polytopes,  developments of  sparse representations of high-dimensional functions, developments of  kernel-based approximation methods that approximate high-dimensional datasets, to name just a few. Almost all these important contributions utilize, in one way or another, on various methods and techniques arising from OPEs on regular domains. ***   This proposed program consists of the following two  integrated parts:  (i) Study qualitative and quantitative features of OPEs and CFs  on regular domains; (ii) Explore ways to apply results of Part (i)  to challenges in QA  and other related areas, such as numerical analysis, discrete geometry and convex geometry. In all phases of the research, the  Dunkl theory of weighted OPEs on spheres is expected to be the useful tool. The research will develop new construction methods for positive CFs on spheres and related domains and will enhance our understanding of how geometry of the underlying domain influences the quality of high dimensional approximation. It will also stimulate interest in students and provide them with a greater opportunity to learn the fundamentals and powerful techniques of different disciplines, and their interrelations. Results of the proposed research  will have potential applications in a number of areas, such as numerical analysis, statistics, geometric modeling, geophysics, differential equations and computing, and imaging and information  technologies.  **