Mathematical Sciences: Gaussian Cubature Formula and its Applications

徐9302721 本计画是关于特殊函数理论在求积问题上的应用。 求积是积分方法的术语，尤其是数值积分。 它已经在一维高度发达的工具，如高斯求积公式。 本工作旨在通过容积法在几个维度上进行数值积分。 这种概括远非直截了当。 求积式和求三次式需要正交多项式的知识，特别是具有公共根的知识。 在多维情况下，具有共同根的正交多项式被认为是罕见的。 然而，最近的工作现在确定有许多高斯立方。 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。 确立了若干目标。 首先，得到高斯立方的适用特征，然后构造新的有效的数值积分公式。 工作也将完成应用的容积公式的理论插值在几个变量。 正交多项式在数值积分中的使用提供了在对于最高可能次数的多项式精确的意义上是最佳的公式。 因此，他们提供了良好的近似积分的光滑函数。 最近的突破进行这些想法在几个变量提供了一个强大的多元cubature和近似理论在几个变量，一个主题仍然处于起步阶段的可能性。 ***

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. ***