Mathematical Sciences: Gaussian Cubature Formula and its Applications

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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. ***