Error analysis in some Gauss–Turán–Radau and Gauss–Turán–Lobatto quadratures for analytic functions

Abstract

We consider the generalized Gauss–Turán quadrature formulae of Radau and Lobatto type for approximating ∫ −1 1 f(t)w(t) dt . The aim of this paper is to analyze the remainder term in the case when f is an analytic function in some region of the complex plane containing the interval [−1,1] in its interior. The remainder term is presented in the form of a contour integral over confocal ellipses (cf. SIAM J. Numer. Anal. 80 (1983) 1170). Sufficient conditions on the convergence for some of such quadratures, associated with the generalized Chebyshev weight functions, are found. Using some ideas from Hunter (BIT 35 (1995) 64) we obtain new estimates of the remainder term, which are very exact. Some numerical results and illustrations are shown.