An Error Expansion for some Gauss–Turán Quadratures and L1-Estimates of the Remainder Term

Abstract

Our aim in this paper is to obtain error expansions in the Gauss–Turan quadrature formula ∫−11f(t)w(t) dt=∑ν=1n∑i=02sAi,νf(i)(τν)+Rn,s(f), in the case when f is an analytic function in some region of the complex plane containing the interval [−1,1] in its interior. Using a representation of the remainder term Rn,s(f) in the form of contour integral over confocal ellipses, we obtain Rn,1(f) for the four Chebyshev weights and Rn,2(f) for the Chebyshev weight of the first kind. Also, we get a few new L1-estimates of the remainder term, which are stronger than the previous ones. Some numerical results, illustrations and comparisons are also given.