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。

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.