$L^p$-Estimation for the gaussian quadratures and numerical application

Abstract

Gauss-type quadratures formulae (Gaussian quadrature, Gauss-Radau rule and Gauss-Lobatto rule) for analytic functions with Chebyshev weight functions, have been known for a long time. Recently, many authors [20][23] considered the Gaussian quadrature formulae for the Bernstein-Szego weight functions. Kaneko and Xu [14] established that application of the quadrature scheme yields numerical solutions of the weakly singular Fredholm integral equation of the second kind. For analytic functions it is well-known that the remainder term can be represented by the contour integral. In this work we present the $L^2$-estimation for the kernel $K_n$ of the remainder term with respect to one of four Chebyshev weight functions and the error bound of the type on the elliptic contour ◁수식 삽입▷(원문을 참조하세요) where l(Γ) denotes the length of the contour Γ and give an expression for the kernel $K_n$ of the remainder terms for Gauss-Radau and Gauss-Lobatto rules with end points of multiplicity r using the Laurent series expansion. Also we prove the convergence of kernel $K_n$ and obtain the error bound of the type ◁수식 삽입▷(원문을 참조하세요) on the circular and elliptic contours.