The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type

Abstract

We consider the Gauss-Kronrod quadrature formulae for the Bernstein-SzegoIi weight functions consisting of any one of the four Chebyshev weights divided by the polynomial . For analytic functions, the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points a" 1 and sum of semi-axes rho > 1, for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed by S. Notaris (Numer. Math. 103, 99-127, 2006).