In this paper we consider error bounds of the Clenshaw–Curtis and related quadrature formulae, with respect to the Legendre weight function on the interval $[-1,1]$, for an analytic integrand $f$. For certain spaces of analytic functions, Notaris [12] derived estimates for the Clenshaw–Curtis, the Basu, and the
Fejér quadrature formula of the first kind. Inspired by these results we derive another kind of error bounds. As
is well known, in the case of analytic integrands, the error of such a quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel of the mentioned quadrature formulae on elliptic contours with foci at the points $\pm 1$ and the sum of semi-axes $\rho>1$ and derive some error bounds. In addition, we obtain a result about the behavior of the modulus of the corresponding kernels on those ellipses in certain cases. Numerical examples demonstrating the accuracy of such error bounds are included.
Keywords: Clenshaw–Curtis quadrature, Basu quadrature, Fejér quadrature of the first kind, error bounds, analytic integrand, ellipse
