Gaussian versus Optimal Integration of Analytic Functions

We consider error estimates for optimal and Gaussian quadrature for-mulae if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szegg o-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulae are close to the obtained lower bounds, which proves the quality of the Gaussian formulae and also of the lower bounds. In the sequel, diierent regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulae are near-optimal. For classes of simply connected regions of an-alyticity, which are additionally symmetric to the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulae and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christooel-function for the constant weight function and arguments outside the interval of integration.