Cubature Formulas, Orthogonal Expansions and Quantitative Approximation on Regular Domains

正则域上的体积公式、正交展开和定量逼近

• 批准号: RGPIN-2015-04702
• 负责人: Dai, Feng
• 金额: $ 1.82万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=589960
• 关键词: Cubature; Formulas; Orthogonal; Expansions; Quantitative

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.

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。