On the remainder term of Gauss–Radau quadrature with Chebyshev weight of the third kind for analytic functions

Abstract

For analytic functions the remainder term of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points ∓1 and a sum of semi-axes ρ>1, for Gauss–Radau quadrature formula with Chebyshev weight function of the third kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi in his paper [W. Gautschi, On the remainder term for analytic functions of Gauss–Lobatto and Gauss–Radau quadratures, Rocky Mounatin J. Math. 21 (1991) 209-206]. In this way the last unproved conjecture from the mentioned paper is now verified.