Mathematical Sciences: Gaussian Cubature Formula and its Applications

徐9302721这个项目是关于特殊函数理论在求积问题中的应用。求积是指积分方法，尤指数值积分方法。利用诸如高斯求积公式等工具，它在一维上得到了高度发展。目前的工作试图通过多个维度的体积方法进行数值积分。这种概括远非直截了当。求积和立方需要了解正交多项式，特别是具有公根的知识。在多维情况下，具有公共根的正交多项式被认为是罕见的。然而，最近的研究表明，现在有许多高斯型孵化器。这项工作的目的是使用新开发的工具来进一步研究这一主题。确立了几个目标。首先得到高斯立方的适用刻画，然后构造新的有效的数值积分公式。我们还将把体积公式应用到多变量插补理论中。在数值积分中使用正交多项式所提供的公式在对最高可能次数的多项式精确的意义上是最优的。因此，它们可以很好地逼近光滑函数的积分。最近在将这些概念推广到多个变量方面的突破表明，在多个变量中建立强大的多变量容度和近似理论是可能的，这是一个仍处于起步阶段的学科。***

Xu 9302721 This project concerns applications of the theory of special functions to problems of quadrature. Quadrature is a term given to methods of integration, especially numerical. It has been highly developed in one dimension with tools such as the Gaussian quadrature formula. The present work seeks to carry out numerical integration through methods of cubature in several dimensions. The generalization is far from straightforward. The quadratures and cubatures require knowledge of orthogonal polynomials, particularly with common roots. In the multidimensional case, orthogonal polynomials with common roots were thought to be rare. However recent work now establishes that there are many Gaussian cubatures. The purpose of this work is to use the newly developed tools to investigate this topic further. Several goals are established. First, to obtain applicable characterization of Gaussian cubatures, then to construct new efficient numerical integration formulae. Work will also be done applying the cubature formulae to the theory of interpolation in several variables. The use of orthogonal polynomials in numerical integration provides formulas which are optimal in the sense of being exact for polynomials of the highest possible degree. As such they provide good approximations to integrals of smooth functions. The recent breakthroughs in carrying these ideas over to several variables holds out the possibility of a robust multivariate cubature and approximation theory in several variables, a subject still in its infancy. ***