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逼近一般的、可能复杂的函数，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该近似中的误差大小。立方体公式(CFS)和正交多项式展开式(OPE)在质量保证和许多其他相关领域中发挥着至关重要的作用，如数值积分、计算机层析成像、编码理论、数据拟合和压缩传感。Cf本身对于高维积分的实际评估是必不可少的，而opes已经成为研究和构建高维积分的主要工具。在工程、金融、生物、医学和量子化学应用的驱动下，CFS、OPE和QA中的现代问题往往在各种规则领域(如球体、单纯形、球体和立方体)上以几个变量表示。 但是，最近在正规域名上的CFS、OPE和QA的各个方向上都做出了很多非常重要的贡献。这些贡献包括：解决了长期存在的关于球面设计的最佳尺寸的Korevaar-Meyer猜想，在Borsuk猜想的反例中创造了新的世界纪录，球面上的点的普遍最优分布，球、球面和多面体上的光滑方面的多项式逼近速度的刻画，高维函数的稀疏表示的发展，基于核的逼近方法的发展，以及近似高维数据集的方法，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来自常规领域上的OPE的各种方法和技术。 这一拟议的计划包括以下两个完整的部分：(I)研究OPES和CFS在规则域上的定性和定量特征；(Ii)探索如何更好地应用第(I)部分的结果来应对QA领域和其他相关领域的挑战，如数值分析、离散几何和凸几何。在这项研究的所有阶段，邓克尔关于球体上的加权Opes理论有望成为有用的工具。这项研究将开发球面及相关区域上正向CFS的新构造方法，并将增强我们对基本区域的几何形状如何影响高维空间近似的质量的理解。它还将激发学生的兴趣，并为他们提供更大的机会，让他们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。这些拟议的研究成果将在多个领域具有潜在的应用，如数值分析、统计、几何建模、地球物理、微分方程和计算，以及成像和信息技术。

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.