Ruymán Cruz-Barroso

On bi-orthogonal systems of trigonometric functions and quadrature formulas for periodic integrands

In this paper, quadrature formulas with an arbitrary number of nodes and exactly integrating trigonometric polynomials up to degree as high as possible are constructed in order to approximate 2π-periodic weighted integrals. For this purpose, certain bi-orthogonal systems of trigonometric functions are introduced and their most relevant properties studied. Some illustrative numerical examples are also given. The paper completes the results previously given by Szegő in Magy Tud Akad Mat Kut Intez Közl 8:255–273, 1963 and by some of the authors in Annales Mathematicae et Informaticae 32:5–44, 2005.

Rational Szego quadratures associated with Chebyshev weight functions

In this paper we characterize rational Szegýo quadrature formulas associated with Chebyshev weight functions, by giving explicit expressions for the corresponding para-orthogonal rational functions and weights in the quadratures. As an application, we give characterizations for Szegýo quadrature formulas associated with rational modifications of Chebyshev weight functions. Some numerical experiments are finally presented.

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.

Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these ORF when the poles are all outside or all inside the unit disk, or when they can be anywhere in the extended complex plane outside the unit circle. Some properties of matrices that are the product of elementary unitary transformations will be proved and some connections with related algorithms for direct and inverse eigenvalue problems will be explained.

Multiple orthogonal polynomials on the unit circle. Normality and recurrence relations

Multiple orthogonal polynomials on the unit circle (MOPUC) were introduced by J. Minguez and W. Van Assche for the first time in 2008. Some applications were given there and recurrence relations were obtained from a Riemann-Hilbert problem.This paper is a second contribution to this field. We first obtain a determinantal formula for MOPUC (multiple Heines formula) and we analyze the concept of normality, from a dynamical point of view and by presenting a first example: the combination of the Lebesgue and Rogers-Szeg? measures. Secondly, we deduce recurrence relations for MOPUC without using Riemann-Hilbert analysis, only by considering orthogonality conditions. This new approach allows us to complete the recurrence relations in the situation when the origin is a zero of MOPUC, a case that was not considered before. As a consequence, we give an appropriate definition of multiple Verblunsky coefficients. A multiple version of the well known Szeg? recurrence relation is also obtained. Here, the coefficients that appear in the recurrence satisfy certain partial difference equations that are used to present a recursive algorithm for the computation of MOPUC. A discussion on the Riemann-Hilbert approach that also includes the case when the origin is a zero of MOPUC is presented. Some conclusions and open questions are finally mentioned.

Matrix methods for quadrature formulas on the unit circle. A survey

In this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle.Like nodes and weights of Gauss quadrature formulas (for the estimation of integrals with respect to measures on the real line) can be computed from the eigenvalue decomposition of the Jacobi matrix, Szeg? quadrature formulas (for the approximation of integrals with respect to measures on the unit circle) can be obtained from certain unitary five-diagonal or unitary Hessenberg matrices that characterize the recurrence for an orthogonal (Laurent) polynomial basis. These quadratures are exact in a maximal space of Laurent polynomials.Orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles. More general Szeg? quadrature formulas can be obtained that are exact in certain spaces of rational functions. In this context, the nodes and the weights of these rules are computed from the eigenvalue decomposition of an operator Mobius transform of the same five-diagonal or Hessenberg matrices.