Volume 65, pp. 308-325, 2026.

Error bounds of Clenshaw-Curtis and related quadratures for functions analytic on ellipses

A. V. Pejčev and M. M. Spalević

Abstract

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.

Full Text (PDF) [316 KB], BibTeX , DOI: 10.1553/etna_vol65s308

Key words

Clenshaw–Curtis quadrature, Basu quadrature, Fejér quadrature of the first kind, error bounds, analytic integrand, ellipse

AMS subject classifications

65D30, 65D32

Links to the cited ETNA articles

[3]	Vol. 53 (2020), pp. 352-382 D. Lj. Djukić, R. M. Mutavdžić Djukić, A. V. Pejčev, and M. M. Spalević: Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results
[6]	Vol. 55 (2022), pp. 424-437 D. R. Jandrlić, D. M. Krtinić, Lj. V. Mihić, A. V. Pejčev, and M. M. Spalević: Error bounds for Gaussian quadrature formulae with Legendre weight function for analytic integrands