Characterizing the measures on the unit circle with exact quadrature formulas in the space of polynomials

Abstract

a b s t r a c tIn the present paper we characterize the measures on the unit circle for which there exists a quadrature formula with a fixed number of nodes and weights and such that it exactly integrates all the polynomials with complex coefficients. As an application we obtain quadrature rules for polynomial modifications of the Bernstein measures on [−1, 1], having a fixed number of nodes and quadrature coefficients and such that they exactly integrate all the polynomials with real coefficients.

Chebyshev series method for computing weighted quadrature formulas

“…It is clear that W is a generic weight that includes the four classical Chebyshev weight functions. It is used a technique similar to that by Berriochoa et al [1], to prove an exact formula for λ n,k (KW/q). Unlike the method proposed by Clenshaw and Curtis [4], here we only use the Chebyshev series expansion of a factor K.…”