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

16 pages, no figures.-- MSC2000 codes: 33C47; 42C05.MR#: MR1988489 (2004d:42042)Zbl#: Zbl 1032.42030In 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–Szegö measure and the Lebesgue measure. In this case we obtain more simple explicit algebraic relations as well as the relation between the corresponding Szegö’s functions.First (A.C.) and third (C.P.) author's research was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant number BFM2000-0015 and by Universidad de Vigo and Xunta de Galicia. Second author (F.M.)'s research was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM2000-0206-C04-01 and INTAS project INTAS2000-272.Publicad

A study of the generalized Christoffel functions with applications

10 pages, no figures.-- MSC2000 code: 42C05.MR#: MR1803313 (2001m:42044)Zbl#: Zbl 0965.42015In this paper we solve the problem of minimizing the norm of polynomials of degree less than or equal to n, verifying linear restrictions. This case extends in a natural way a problem studied by Grenander and Rosenblatt. We obtain algebraic properties of the solution which enables us to compute it and we present some applications.The research of the first author (E.B.) was supported by Dirección General de Enseñanza Superior (DGES) of Spain PB96-0344. The research of the second author (A.C.) was supported by Dirección General de Enseñanza Superior (DGES) of Spain PB96-0344. The research of the third author (F.M.) was supported by Dirección General de Enseñanza Superior (DGES) of Spain PB96-0120-C03-01.Publicad

Second kind functionals for the Laguerre-Hahn affine class on the unit circle

16 pages, no figures.-- MSC1991 codes: 33C47, 42C05.MR#: MR2032331 (2004j:33013)Zbl#: Zbl 1036.42024The aim of this paper is the study of some transformations in the Laguerre-Hahn affine class that do not preserve the class. Under very general conditions, we establish that the second kind functional associated with a Laguerre-Hahn affine functional does not belong to the Laguerre-Hahn affine class. The transformations related to the associated polynomials and quadratic decomposition of a sequence of orthogonal polynomials are also considered.The research of first (A.C.) and third (C.P.) authors was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant number BFM2000-0015 and by Universidad de Vigo and Xunta de Galicia. The research of second author (F.M.) was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM2000-0206-C04-01 and INTAS project INTAS2000-272.Publicad

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

24 pages, no figures.-- MSC2000 codes: 33C47, 42C05.-- Dedicated to Professor Dr. Mariano Gasca with occasion of his 60th anniversary.MR#: MR2350346 (2008m:33032)Zbl#: Zbl 1109.33010In 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.The research was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant number BFM2000-0015, as well as BFM2003-06335-C03-C02.Publicad

A new numerical quadrature formula on the unit circle

11 pages, no figures.-- MSC2000 codes: 33C47, 42C05, 65D30.MR#: MR2335810 (2008i:33038)Zbl#: Zbl 1126.65023In this paper we study a quadrature formula for Bernstein–Szegö measures on the unit circle with a fixed number of nodes and unlimited exactness. Taking into account that the Bernstein–Szegö 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.This work was supported by Ministerio de Educación y Ciencia under grants number MTM2005-01320 (E. B. and A. C.) and MTM2006-13000-C03-02 (F. M.).Publicad

A scalar Riemann boundary value problem approach to orthogonal polynomials on the circle

8 pages, no figures.-- MSC2000 codes: 33C47; 42C05.MR#: MR2252097 (2007k:33010)Zbl#: Zbl 1130.42025A scalar Riemann boundary value problem defining orthogonal polynomials on the unit circle and the corresponding functions of the second kind is obtained. The Riemann problem is used for the asymptotic analysis of the polynomials orthogonal with respect to an analytical real-valued weight on the circle.The research was supported by INTAS Research Network NeCCA 03-51-6637. The first author was also supported by the Grants RFBR 05-01-00522, NSh-1551.2003.1 and by the program N1 DMS, RAS. The second authorwas supported by Ministerio de Ciencia y Tecnología under Grant number MTM2005-01320. The third author was supported by Ministerio de Ciencia y Tecnología under Grant number BFM2003-06335-C03-02.Publicad

Szegő transformation and zeros of analytic perturbations of Chebyshev weights

In this contribution we present a method to find the closest zeros to ±1 for orthogonal polynomials with respect to analytic perturbations of the Chebyshev weight functions in [−1,1]. The error order we obtain is O(n−6)Ministerio de Economía y Competitividad | Ref. AGL2014-60412-RMinisterio de Economía y Competitividad | Ref. MTM2015-65888-C4-2-

New quadrature rules for Bernstein measures on the interval [-1,1]

13 pages, no figures.-- MSC2000 codes: 33C47, 42C05.MR#: MR2480082Zbl#: Zbl pre05602092In the present paper, we obtain quadrature rules for Bernstein measures on [-1, 1], having a fixed number of nodes and weights such that they exactly integrate functions in the linear space of polynomials with real coefficients.The first three authors were partially supported by Ministerio de Educación y Ciencia under grant number MTM2005-01320. The fourth author was partially supported by Ministerio de Educación y Ciencia under grant number MTM2006-13000-C03-02 and project CCG07-UC3M/ESP-3339 with the financial support of Comunidad de Madrid and Universidad Carlos III de Madrid.Publicad

Gibbs–Wilbraham phenomenon on Lagrange interpolation based on analytic weights on the unit circle

This paper is devoted to study Lagrange interpolation based on nodal systems constituted by the roots of para-orthogonal polynomials with respect to analytic weights on the unit circle. The presented results address, in addition to algorithmic and convergence questions for continuous and discontinuous functions, a detailed study of the Gibbs-Wilbraham phenomenon.Ministerio de Economía y Competitividad | Ref. AGL2014-60412-

Gibbs–Wilbraham oscillation related to an Hermite interpolation problem on the unit circle

The aim of this piece of work is to study some topics related to an Hermite interpolation problem on the unit circle. We consider as nodal points the zeros of the para-orthogonal polynomials with respect to a measure in the Baxter class and such that the sequence of the first derivative of the reciprocal of the orthogonal polynomials is uniformly bounded on the unit circle. We study the convergence of the Hermite–Fejér interpolants related to piecewise continuous functions and we describe the sets in which the interpolants uniformly converge to the piecewise continuous function as well as the oscillatory behavior of the interpolants near the discontinuities, where a Gibbs–Wilbraham phenomenon appears. Finally we present some numerical experiments applying the main results and by considering nodal systems of interest in the theory of orthogonal polynomials.Ministerio de Economía y Competitividad | Ref. AGL 2014-60412-