A note on the bounds of the error of Gauss–Turán-type quadratures

摘要

This note is concerned with estimates for the remainder term of the Gauss–Turán quadrature formula,Rn,s(f)=∫-11w(t)f(t)dt-∑ν=1n∑i=02sAi,νf(i)(τν),where w(t)=(Un-1(t)/n)21-t2 is the Gori–Michelli weight function, with Un-1(t) denoting the (n-1)th degree Chebyshev polynomial of the second kind, and f is a function analytic in the interior of and continuous on the boundary of an ellipse with foci at the points ±1 and sum of semiaxes ϱ>1. The present paper generalizes the results in [G.V. Milovanović, M.M. Spalević, Bounds of the error of Gauss–Turán-type quadratures, J. Comput. Appl. Math. 178 (2005) 333–346], which is concerned with the same problem when s=1.