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.

期望正交规则是内部的，即规则的所有节点都位于度量支持的凸包中。然后可以将该规则应用于具有接近测度支持的凸包的奇点的函数的近似积分。本文研究了第一类修正切比雪夫测度的广义平均高斯求积公式是否是内部的。这些规则用于估计与第一类修正切比雪夫测度相关的高斯正交规则中的误差。能够评估正交规则中的误差以便能够选择一个规则来提供足够精度的所需积分的近似值而不必在不必要的许多节点上评估被积函数是非常有趣的。发现所考虑的一些广义平均高斯求积公式不是内部的。我们将证明这些规则的一些截断变体是内部的，因此当被积函数在接近积分区间的实轴上具有奇点时，也可以应用于估计高斯正交规则中的误差。