A. Cachafeiro

A new numerical quadrature formula on the unit circle

In this paper we study a quadrature formula for Bernstein–Szegő measures on the unit circle with a fixed number of nodes and unlimited exactness. Taking into account that the Bernstein–Szegő measures are very suitable for approximating an important class of measures we also present a quadrature formula for this type of measures such that the error can be controlled with a well-bounded formula.

Algorithms for solving Hermite interpolation problems using the Fast Fourier Transform

We present a method for computing the Hermite interpolation polynomial based on equally spaced nodes on the unit circle with an arbitrary number of derivatives in the case of algebraic and Laurent polynomials. It is an adaptation of the method of the Fast Fourier Transform (FFT) for this type of problems with the following characteristics: easy computation, small number of operations and easy implementation. In the second part of the paper we adapt the algorithm for computing the Hermite interpolation polynomial based on the nodes of the Tchebycheff polynomials and we also study Hermite trigonometric interpolation problems.

Lebesgue perturbation of a quasi-definite Hermitian functional. The positive definite case

Abstract In this work we study the problem of orthogonality with respect to a sum of measures or functionals. First we consider the case where one of the functionals is arbitrary and quasi-definite and the other one is the Lebesgue normalized functional. Next we study the sum of two positive measures. The first one is arbitrary and the second one is the Lebesgue normalized measure and we obtain some relevant properties concerning the new measure. Finally we consider the sum of a Bernstein–Szegő measure and the Lebesgue measure. In this case we obtain more simple explicit algebraic relations as well as the relation between the corresponding Szegő’s functions.

Some improvements to the Hermite-Fejér interpolation on the circle and bounded interval

In this paper, we study the convergence of the Hermite-Fejer and the Hermite interpolation polynomials, which are constructed by taking equally spaced nodes on the unit circle. The results that we obtain are concerned with the behaviour outside and inside the unit circle, when we consider analytic functions on a suitable domain. As a consequence, we achieve some improvements on Hermite interpolation problems on the real line. Since the Hermite-Fejer and the Hermite interpolation problems on [-1,1], with nodal systems mainly based on sets of zeros of orthogonal polynomials, have been widely studied, in our contribution we develop a theory for three special nodal systems. They are constituted by the zeros of the Tchebychef polynomial of the second kind joint with the extremal points -1 and 1, the zeros of the Tchebychef polynomial of the fourth kind joint with the point -1, and the zeros of the Tchebychef polynomial of the third kind joint with the point 1. We present a simple and efficient method to compute these interpolation polynomials and we study the convergence properties.

A family of Sobolev orthogonal polynomials on the unit circle

Abstract The aim of this paper is to study the polynomials orthogonal with respect to the following Sobolev inner product: 〈P,Q〉 s 1 = ∫ 0 2π P( e i θ ) Q( e i θ ) d μ(θ)+ 1 λ ∫ 0 2π P′( e i θ ) Q′( e i θ ) d ν(θ), z= e i θ , λ>0, where ν is the normalized Lebesgue measure and μ is a rational modification of ν. In this situation we analyse the algebraic results and the asymptotic behaviour of such orthogonal polynomials. Moreover some properties about the distribution of their zeros are given.

Lebesgue Sobolev orthogonality on the unit circle

This paper is devoted to the study of asymptotic properties of the orthogonal polynomials with respect to a Sobolev inner product 〈f(z),g(z)〉s = ∫02π f(eiθ))g(eiθ)dμ(θ)+∑k=1pλk∫02πfk(eiθ)g(eiθ)dθ2π, z=eiθ, with dμ(θ) a finite positive Borel measure on [0, 2π] with an infinite set as support verifying the Szegő condition, λ1 > 0, λk ⩾ 0 (k = 2,…, p) and dθ2π the normalized Lebesgue measure on [0, 2π]. Our aim is to extend some previous results that we have obtained in [2, 3] when the measure μ belongs to the Bernstein-Szegő class and p = 1.

Gibbs phenomenon in the Hermite interpolation on the circle

Hermite-Fejer interpolation problems on the unit circle and bounded interval are usually studied in relation with continuous functions. There are few references concerning these problems for functions with discontinuities. Thus the aim of this paper is to describe the behavior of the Hermite-Fejer and Hermite interpolants for piecewise continuous functions on the unit circle, analyzing the corresponding Gibbs phenomenon near the discontinuities and providing the asymptotic amplitude of the Gibbs height.

Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure

Abstract In the present paper we study the orthogonal polynomials with respect to a measure which is the sum of a finite positive Borel measure on [0,2π] and a Bernstein–Szegö measure. We prove that the measure sum belongs to the Szegö class and we obtain several properties about the norms of the orthogonal polynomials, as well as, about the coefficients of the expression which relates the new orthogonal polynomials with the Bernstein–Szegö polynomials. When the Bernstein–Szegö measure corresponds to a polynomial of degree one, we give a nice explicit algebraic expression for the new orthogonal polynomials.

Strong asymptotics inside the unit disk for Sobolev orthogonal polynomials

Abstract In the present paper, we give sufficient conditions in order to establish the extension of the strong asymptotics up to the boundary and inside the unit disk for Sobolev orthogonal polynomials. We consider the following Sobolev inner product on the unit circle: with μ0 a finite positive Borel measure on [0,2π] and μ1 a measure in the Szegős class. On the assumption that the Caratheodory function of μ0 and the Szegő function ofμ1 have analytic extension, we prove that the asymptotic formula holds true outside the disk and it can be extended inside the disk.

Algorithms and convergence for Hermite interpolation based on extended Chebyshev nodal systems

The Chebyshev nodal systems play an important role in the theory of Hermite interpolation on the interval - 1 , 1 ] . For the cases of nodal points corresponding to the Chebyshev polynomials of the second kind U n ( x ) , the third kind V n ( x ) and the fourth kind W n ( x ) , it is usual to consider the extended systems, that is, to add the endpoints -1 and 1 to the nodal system related to U n ( x ) , to add -1 to the nodal system related to V n ( x ) and to add 1 to the nodal system related to W n ( x ) . The interpolation methods that are usually used in connection with these extended nodal systems are quasi-Hermite interpolation and extended Hermite interpolation, and it is well known that the performance of these two great methods is quite good when it comes to continuous functions.This work attempts to complete the theory concerning these extended Chebyshev nodal systems. For this, we have obtained a new formulation for the Hermite interpolation polynomials based upon barycentric formulas. The feature of this approach is that the derivatives of the function at the endpoints of the interval are also prescribed. Further, some convergence results are obtained for these extended interpolants when apply to continuous functions.