Approximation in weighted spaces on regular domains
DescriptionIn mathematics, it is often desirable to approximate general functions on various compact domains by simpler functions. Then one naturally asks how good these approximations are, and what are the best rates that one can theoretically achieve for a given class of functions. For studying these problems, cubature formulas (CFs) and orthogonal polynomial expansions (OPEs) have been playing vital roles in dealing with various important topics, such as n-widths, spherical wavelet frames, distribution of points on spheres, covering and packing problems, coding theory and some aspects of convex geometry.Modern problems in CFs, OPEs are often formulated in several variables on various regular domains, such as spheres, cubes, balls, and simplexes, driven by applications in engineering, biology, medicine and quantum physics. At the current stage, however, in contrast to the situation in one variable, many fundamental problems have not been solved. The purpose of this project is to understand the qualitative and quantitative features of CFs and weighted OPEs (WOPEs) on regular domains with weights invariant under certain reflection group. Our topics will be focused on: Convergence of Cesáro means in weighted Lp spaces; Optimal Chebyshev-type cubature formulas in weighted Besov spaces. As long-term goals, we shall apply the expected results to challenges in quantitative approximation and other related areas, such as functional analysis, discrete geometry and convex geometry.
|Effective start/end date||1/01/20 → …|