Anti-Gaussian quadrature rules related to multiple orthogonal polynomials

Abstract

Multiple orthogonal polynomials represent one of the generalizations of orthogonal polynomials, in the sense that they satisfy orthogonality with respect to r different weight functions simultaneously. Anti-Gaussian quadrature formulas on the space of algebraic polynomials were introduced in 1996 by Laurie ([1]). These quadrature formulas have the property that their error is equal in magnitude but of opposite sign to the corresponding Gaussian quadrature rules. Here, we analyze a set of anti-Gaussian quadrature rules for the optimal set of quadrature rules in Borges' sense (see [2]), which refers to the observed multiply orthogonal polynomials, and define a set of averaged quadrature formulas.