Francisco Perdomo-Pío

Rational approximants associated with measures supported on the unit circle and the real line

Abstract As a continuation of the well known connection between the theory of orthogonal polynomials on the unit circle and the interval [ − 1 , 1 ] , in this paper properties concerning error and convergence of certain rational approximants associated with the measures d μ ( t ) and d σ ( θ ) = | d μ ( cos θ ) | supported on [ − 1 , 1 ] and the unit circle respectively are deduced. Numerical illustrations are also given.

An application of Szegő polynomials to the computation of certain weighted integrals on the real line

In this paper, a new approach in the estimation of weighted integrals of periodic functions on unbounded intervals of the real line is presented by considering an associated weight function on the unit circle and making use of both Szegő and interpolatory type quadrature formulas. Upper bounds for the estimation of the error are considered along with some examples and applications related to the Rogers-Szegő polynomials, the evaluation of the Weierstrass operator, the Poisson kernel and certain strong Stieltjes weight functions. Several numerical experiments are finally carried out.

A connection between Szegő-Lobatto and quasi Gauss-type quadrature formulas

In this paper we obtain new results on positive quadrature formulas with prescribed nodes for the approximation of integrals with respect to a positive measure supported on the unit circle.We revise Szeg?-Lobatto rules and we present a characterization of their existence. In particular, when the measure on the unit circle is symmetric, this characterization can be used to recover, in a more elementary way, a recent characterization result for the existence of positive quasi Gauss, quasi Radau and quasi Lobatto rules (quasi Gauss-type), due to B. Beckermann et. al. Some illustrative numerical examples are finally carried out in order to show the powerfulness of our results.