Error bounds for Gauss-type quadratures with Bernstein-Szego weights

Abstract

The paper is concerned with the derivation of error bounds for Gauss-type quadratures with Bernstein-Szego weights, integral(1)(-1) f(t)w(t) dt = G(n)[f] + R-n(f), G(n)[f] = Sigma(n)(nu=1) lambda(nu)f(tau(nu)) (n is an element of N), where f is an analytic function inside an elliptical contour epsilon(rho) with foci at -/+ 1 and sum of semi-axes rho > 1, and w is a nonnegative and integrable weight function of Bernstein-Szego type. The derivation of effective bounds on vertical bar R-n(f)vertical bar is possible if good estimates of max(z is an element of epsilon rho) vertical bar K-n(z)vertical bar are available, especially if one knows the location of the extremal point eta is an element of epsilon(rho) at which vertical bar K-n vertical bar attains its maximum. In such a case, instead of looking for upper bounds on max(z is an element of epsilon rho) vertical bar K-n(z)vertical bar, one can simply try to calculate vertical bar Kn(eta, w)vertical bar. In the case under consideration, i.e. when w(t) = (1 - t(2))(-1/2)/beta(beta - 2 alpha)t(2) + 2 delta(beta - alpha)t + alpha(2) + delta(2), t is an element of (-1, 1), for some alpha, beta, delta, which satisfy 0 lt alpha lt beta, beta not equal 2 alpha, vertical bar delta vertical bar lt beta - alpha, the location on the elliptical contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective bounds on vertical bar R-n(f)vertical bar. The quality of the derived bounds is analyzed by a comparison with other error bounds proposed in the literature for the same class of integrands.