Valentin Kadushkin: Ortogonala polynom och Gauss-kvadratur

Abstract.

This thesis examines the theory of Gaussian quadrature and its connection to orthogonal polynomials. Gaussian quadrature is an efficient numerical method for approximating integrals especially if the integrand can be expressed as a polynomial or is well approximated by one. The method is based on choosing nodes and weights in such a way that the integral is exact for all polynomials up to degree 2n−1. The nodes are zeros of an orthogonal polynomial while the weights are determined based on the orthogonality properties. The thesis first discusses the basic properties of orthogonal polynomials, focusing on classical examples such as Legendre polynomials. This is followed by a formal proof of the exactness of Gaussian quadrature as well as a discussion of the method’s applications. The aim is to provide a theoretical understanding of why Gaussian quadrature works and how orthogonal polynomials play a central role in its construction.