Internality of Two-Measure-Based Generalized Gauss Quadrature Rules for Modified Chebyshev Measures II

Gaussian quadrature rules are commonly used to approximate integrals with respect to a non-negative measure dσ^. It is important to be able to estimate the quadrature error in the Gaussian rule used. A common approach to estimating this error is to evaluate another quadrature rule that has more nodes and higher algebraic degree of precision than the Gaussian rule, and use the difference between this rule and the Gaussian rule as an estimate for the error in the latter. This paper considers the situation when dσ^ is a Chebyshev measure that is modified by a linear factor and a linear divisor, and investigates whether the rules in a recently proposed new class of quadrature rules for estimating the error in Gaussian rules are internal, i.e., if all nodes of the new quadrature rules are in the interval (−1,1). These new rules are defined by two measures, one of which is a modified Chebyshev measure dσ^. The other measure is auxiliary.