Optimal quadrature for analytic functions

Abstract

Let G be a domain in the complex plane, which is symmetric with respect to the real axis and contains [−1,1]. For a measure τ on [−1,1] satisfying a regularity condition, we determine the geometric rate of the error of integration, measured uniformly on the class of functions analytic in G and bounded by 1, if the τ-integrals are replaced by optimal interpolatory quadrature formulas with n nodes. We show that this rate is obtained for modified Gauss-quadrature formulas with respect to certain varying weights.