An error expansion for some Gauss-Turan quadratures and L-1-estimates of the remainder term
Our aim in this paper is to obtain error expansions in the Gauss-Turan quadrature formula integral(-1)(1) f(t)w(t) dt = Sigma(nu=1)(n) Sigma(i=0)(2s) A(i,nu) f((i)) (tau(nu)) + R-n,R-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 R-n,R-1(f) in the form of contour integral over confocal ellipses, we obtain R-n,R-1(f) for the four Chebyshev weights and R-n,R-2(f) for the Chebyshev weight of the first kind. Also, we get a few new L-1-estimates of the remainder term, which are stronger than the previous ones. Some numerical results, illustrations and comparisons are also given.
error expansion weight gauss-tunin quadrature s-orthogonal polynomials zeros multiple nodes remainder term for analytic functions contour integral representation error estimate