Error bounds of gaussian quadrature formulae for one class of bernstein-szego weights

Abstract

The kernels K-n(z) in the remainder terms R-n(f) of the Gaussian quadrature formulae for analytic functions f inside elliptical contours with foci at -/+ 1 and a sum of semi- axes rho > 1, when the weight function w is of Bernstein-Szego type w(t) equivalent to w gamma((-1/2,1/2)) (t) = root 1+t/1-t.1/1-4 gamma/(1+gamma)(2) t(2), t is an element of (-1,1), gamma is an element of (-1,0), are studied. Sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with the positive real semi-axis. This leads to effective error bounds of the corresponding Gauss quadratures. The quality of the derived bounds is analyzed by a comparison with other error bounds intended for the same class of integrands. In part our analysis is based on the well-known Cardano formulas, which are not very popular among mathematicians.