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.