Bounds of the error of Gauss–Turán-type quadratures

Abstract

We consider the remainder term of the Gauss–Turán quadrature formulae R n , s ( f ) = ∫ - 1 1 w ( t ) f ( t ) d t - ∑ ν = 1 n ∑ i = 0 2 s A i , ν f ( i ) ( τ ν ) 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.