The error bounds of Gauss-Radau quadrature formulae with Bernstein-SzegA weight functions

Abstract

We consider the Gauss-Radau quadrature formulae integral(1)(-1) f(t)w(t)dt = (n)Sigma(nu=1) lambda(nu)f(tau(nu)) + lambda(n+1) f(c) + R-n(f), with or , for the Bernstein-SzegA weight functions consisting of anyone 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 and a sum of semi-axes , 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 recently by Notaris (Math Comp 10.1090/mcom/2944, 2015).