The error bounds of Gauss quadrature formulae for the modified weight functions of Chebyshev type

TL;DR

This paper analyzes the error bounds of Gauss quadrature formulas for modified Chebyshev weights, deriving new bounds and studying kernel behavior on elliptic contours, supported by numerical validation.

Contribution

It provides new error bounds for Gauss quadrature with modified Chebyshev weights and investigates kernel behavior on elliptic contours, a novel aspect in this context.

Findings

Derived explicit error bounds for quadrature formulas

Analyzed the modulus of kernels on elliptic contours

Validated bounds with numerical examples

Abstract

In this paper, we consider the Gauss quadrature formulae corresponding to some modifications of anyone of the four Chebyshev weights, considered by Gautschi and Li in \cite{gauli}. As it is well known, in the case of analytic integrands, the error of these quadrature formulas can be represented as a contour integral with a complex kernel. We study the kernel, as it is often considered, on elliptic contours with foci at the points $\mp 1$ and such that the sum of semi-axes is $\rho>1$, of the mentioned quadrature formulas, and derive some error bounds for them. In addition, we obtain, for the first time as far as we know, a result about the behavior of the modulus of the corresponding kernels on those ellipses in some cases. Numerical examples checking the accuracy of such error bounds are included.