Internality of averaged gauss quadrature rules for certain modification of jacobi measures

The internality of quadrature rules, i.e., the property that all nodes lie in the interior of the convex hull of the support of the measure, is important in applications, because this allows the application of these quadrature rules to the approximation of integrals with integrands that are defined in the convex hull of the support of the measure only. It is known that the averaged Gauss and optimal averaged Gauss quadrature rules with respect to the four Chebyshev measures modified by a linear divisor are internal. This paper investigates the internality of similarly modified Jacobi measures, namely measures defined by weight functions of the formsw(x) = 1/ (z - x)(1 - x)(a)(1 + x)(b) or w(x) = (z - x)(1 - x)(a)(1 + x)(b) with a, b > -1 and z is an element of R, |z| > 1. We will show that in some cases, depending on the exponents a and b, the averaged and optimal averaged Gauss rules for these measures are internal if the number of nodes is large enough.