Error bound for the gaussian quadrature on the ellipse contour

Abstract

For Gaussian quadrature rules over a finite interval, we develop error bounds from contour integral representation of the remainder term. Here we consider circular and ellipse contours. We attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus. When the contour is a circle, then Gautchi succeeds in answering this question for a large class of weight distributions (including all Jacobi weight). In this case of ellipse contours, we can settle the question for certain Jacobi weight distributions with parameters α = 1/2, β = -1/2. We point out that the kernel of the error functional, at any complex point outside the interval of the integration, can be evaluated accurately.