Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the third and fourth kinds

Gauss quadrature rules are commonly used to approximate integrals determined by a measure with support on a real interval. These rules are known to be internal, i.e., their nodes are in the convex hull of the support of the measure. This allows the application of Gauss rules also when the integrand only is defined on the convex hull of the support of the measure. It is important to be able to estimate the quadrature error that is incurred when using a Gauss rule. Averaged and generalized averaged Gauss quadrature formulas are helpful in this respect. Given an n-node Gauss rule, the associated (2n + 1)-node averaged and generalized averaged Gauss rules are easy to compute. However, they are not guaranteed to be internal, and in this situation they cannot be used for integrands that are defined on the convex hull of the support of the measure only. This paper investigates whether averaged and generalized averaged Gauss quadrature formulas for modified Chebyshev measures of the third and fourth kinds are internal. We show that in situations when this is not the case, truncated variants, that use fewer nodes, are internal. Computed examples that illustrate the performance of the quadrature rules discussed are presented.

Automated summary

This paper investigates the internal property of averaged and generalized averaged Gauss quadrature formulas for modified Chebyshev measures of the third and fourth kinds, and demonstrates that truncated variants have internal property.