Bounds of the error of Gauss–Turán-type quadratures

摘要

We consider the remainder term of the Gauss–Turán quadrature formulaeRn,s(f)=∫-11w(t)f(t)dt-∑ν=1n∑i=02sAi,νf(i)(τν)for analytic functions 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 or circles. A strong error analysis is given for the case with a generalized class of weight functions, introduced recently by Gori and Micchelli. Also, we discuss a general case with an even weight function defined on [-1,1]. Numerical results are included.