The set of anti-Gaussian quadrature rules for the optimal set of quadrature rules for trigonometric polynomials

Abstract

It is well known that the anti-Gaussian quadrature rule, introduced by Laurie in 1996 ([1]), gives the error equal in magnitude but of opposite sign to the error of the corresponding Gaussian quadrature rule. Here, we define a set of anti-Gaussian quadrature rules for the optimal set of quadrature rules in Borges’ sense in the linear space of trigonometric polynomials ([ 2], [ 3]). We consider the orthogonality with respect to the set of r different weight functions, with special attention to even weight functions. Also, we investigate the corresponding class of trigonometric multiple orthogonal polynomials and prove some of their important properties.