Ocene grešaka kvadraturnih formula Gausovog tipa za analitičke funkcije

Abstract

The field of research in this dissertation is concerned with numerical integration,i.e. with the derivation of error bounds for Gauss-type quadratures and their generalizations when we use them to approximate integrals of functions which are analytic inside an elliptical contour Eρ with foci at ∓1 and sum of semi-axes ρ > 1. Special attention is given to Gauss-type quadratures with the special kind of weight functions - weight functions of Bernstein–Szeg˝o type. Three kinds of error bounds are considered in the dissertation, which means analysis of kernels of quadratures, i.e. determination of the location of the extremal point on Eρ at which the modulus of the kernels attains its maximum, calculation of the contour integral of the modulus of the kernel, and, also, series expansion of the kernel. Beyond standard, corresponding quadratures for calculation of Fourier expansion coefficients of an analytic function are also analysed in this dissertation.