A note on an error bound of Gauss-Turán quadrature with the Chebyshev weight

Abstract

In two BIT papers error expansions in the Gauss and Gauss-Turan quadrature formulas with the Chebyshev weight function of the first kind, in the case when integrand is an analytic function in some region of the complex plane containing the interval of integration in its interior, have been obtained. On the basis of that, using a representation of the remainder term in the form of contour integral over confocal ellipses, the upper bound of the modulus of the remainder term, in the cases when certain parameter s (s є N0) takes the specific values s = 0,1,2, has been obtained. Its form for a general s (s є N0) has been supposed in one of the mentioned papers. Here, we prove that formula.