Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the first kind

It is desirable that a quadrature rule be internal, i.e., that all nodes of the rule live in the convex hull of the support of the measure. Then the rule can be applied to approximate integrals of functions that have a singularity close to the convex hull of the support of the measure. This paper investigates whether generalized averaged Gauss quadrature formulas for modified Chebyshev measures of the first kind are internal. These rules are applied to estimate the error in Gauss quadrature rules associated with modified Chebyshev measures of the first kind. It is of considerable interest to be able to assess the error in quadrature rules in order to be able to choose a rule that gives an approximation of the desired integral of sufficient accuracy without having to evaluate the integrand at unnecessarily many nodes. Some of the generalized averaged Gauss quadrature formulas considered are found not to be internal. We will show that some truncated variants of these rules are internal, and therefore can be applied to estimate the error in Gauss quadrature rules also when the integrand has singularities on the real axis close to the interval of integration.