Anti-Gaussian quadrature rule for trigonometric polynomials

Abstract

We investigate an anti-Gaussian quadrature rule with maximal trigonometric degree of exactness with respect to an even weight function on [π, π). Its error is equal in magnitude but of opposite sign to corresponding Gaussian formula. We give the method for its construction based on relations between nodes and weights of the quadrature rule for trigonometric polynomials and those of the quadrature rule for algebraic polynomials. Also, we introduce averaged Gaussian quadrature formula for trigonometric polynomials and, at the end, we give some numerical examples.