Cubature Formulas, Orthogonal Expansions and Quantitative Approximation on Regular Domains

通常希望用更简单、更容易计算的函数来近似一般的、可能复杂的函数，例如代数多项式、多元样条和小波。定量近似（QA）试图尽可能精确地确定近似中的误差大小。体积公式（CF）和正交多项式展开（OPEs）在质量保证和许多其他相关领域（如数值积分、计算机断层扫描、编码理论、数据拟合和压缩感知）中起着至关重要的作用。CF本身对于高维积分的实际计算是必不可少的，而OPEs已经成为研究和构造CF的主要工具。 CF，OPEs和QA中的现代问题通常在各种规则域上的多个变量中进行表述，例如球体，单纯形，球和立方体，由工程，金融，生物学，医学和量子化学中的应用驱动。 近年来，正则域上的CF、OPEs和QA在各个方向上都做出了非常重要的贡献。 这些贡献包括解决了长期存在的Korevaar-Meyer猜想的最佳尺寸的球形设计，新的世界纪录的反例Borsuk猜想，普遍最优分布的点的领域，刻画的速度多项式近似光滑球，球，多面体，发展稀疏表示的高维函数，基于核的近似方法的发展，近似高维数据集，仅举几例。几乎所有这些重要的贡献都以这样或那样的方式利用了从常规域的OPEs中产生的各种方法和技术。 该计划由以下两个部分组成：（i）研究规则域上的OPEs和CF的定性和定量特征;（ii）探索如何将第（i）部分的结果应用于QA和其他相关领域的挑战，如数值分析，离散几何和凸几何。在研究的各个阶段，球上加权运算熵的Dunkl理论有望成为有用的工具。该研究将开发新的建设方法，积极的CF领域和相关领域，并将提高我们的理解如何几何基础域的影响高维近似的质量。它还将激发学生的兴趣，并为他们提供更多的机会来学习不同学科的基础知识和强大的技术，以及它们之间的相互关系。建议的研究结果将有潜在的应用在一些领域，如数值分析，统计，几何建模，物理学，微分方程和计算，成像和信息技术。

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.