Relative Asymptotic of Multiple Orthogonal Polynomials for Nikishin Systems

We prove relative asymptotic for the ratio of two sequences of multiple orthogonal polynomials with respect to Nikishin system of measures. The first Nikishin system ${\mathcal{N}}(\sigma_1,...,\sigma_m)$ is such that for each $k$, $\sigma_k$ has constant sign on its compact support \supp {\sigma_k} \subset \mathbb{R} consisting of an interval $\widetilde{\Delta}_k$, on which $|\sigma_k^{\prime}| > 0$ almost everywhere, and a discrete set without accumulation points in $\mathbb{R} \setminus \widetilde{\Delta}_k$. If {Co}(\supp {\sigma_k}) = \Delta_k denotes the smallest interval containing \supp {\sigma_k}, we assume that $\Delta_k \cap \Delta_{k+1} = \emptyset$, $k=1,...,m-1$. The second Nikishin system ${\mathcal{N}}(r_1\sigma_1,...,r_m\sigma_m)$ is a perturbation of the first by means of rational functions $r_k$, $k=1,...,m,$ whose zeros and poles lie in $\mathbb{C} \setminus \cup_{k=1}^m \Delta_k$.Comment: 30 page

Multipoint rational approximants with preassigned poles

20 pages, no figures.-- MSC1991 codes: 41A21, 42C05, 30E10.MR#: MR1820073 (2002i:41021)Zbl#: Zbl 1160.41305Let $\mu$ be a finite positive Borel measure whose support $S(\mu)$ is a compact regular set contained in $\Bbb R$. For a function of Markov type $\hat\mu(z)=\int_{S(\mu)}d\mu(x)/(z-x)$, z\in\Bbb C\sbs S(\mu), we consider multipoint Padé-type approximants (MPTAs), where some poles are preassigned and interpolation is carried out along a table of points contained in \overline{\Bbb C}\sbs {\rm Co}(S(\mu)) which is symmetrical with respect to the real line. The main purpose of this paper is the study of the `exact rate of convergence' of the MPTAs to the function $\hat\mu$.Research by first author (F.C.) partially carried out at the Mathematics Department of Umeå University under Guest Scholarship from the Swedish Institute. Research by second author (G.L.L.) partially supported by Dirección General de Enseñanza Superior under grant PB 96-0120-CO3-01 and by INTAS under grant 93-0219 EXT.Publicad

Teoría de aproximación

22 págs.-- Publicado en el número monográfico dedicado al XIX Congreso de Ecuaciones Diferenciales Y Aplicaciones (CEDYA), Universidad Carlos III, Leganés (Madrid), Sep 19-23, 2005.Este artículo resume el contenido de las tres conferencias de la sesión monográfica dedicada a la Teoría de Aproximación que se encargó organizar al autor en el marco del XIX Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA2005). A la hora de seleccionar los temas y ponentes, se aplicaron los siguientes principios: que los temas fuesen de actualidad y perspectiva tanto en métodos como en áreas de aplicación y que los ponentes fuesen matemáticos jóvenes españoles que en opinión del autor destacan por sus contribuciones en sus áreas de trabajo. En orden alfabético los ponentes fueron Manuel Bello Hernández, Andrei Martínez Finkelshtein y Xavier Tolsa.Publicad

Ratio asymptotic of Hermite-Padé orthogonal polynomials for Nikishin systems. II

26 pages, no figures.-- MSC2000 codes: Primary 42C05, 30E10; Secondary 41A21.MR#: MR2419380 (2009e:42053)Zbl#: Zbl 1153.42013We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures ${\cal N}(\sigma_1,\dots,\sigma_m)$ such that for each k, σ_k has constant sign on its support consisting on an interval $\tilde\Delta_k$, on which $sigma_k'>0$ almost everywhere, and a set without accumulation points in $\Bbb R\setminus \tilde\Delta_k$.Both authors received support from grants MTM 2006-13000-C03-02 of Ministerio de Ciencia y Tecnología, UC3M-CAM MTM-05-033, and UC3M-CAM CCG-06003M/ESP-0690 of Comunidad Autónoma de Madrid.Publicad

Determining radii of meromorphy via orthogonal polynomials on the unit circle

19 pages, no figures.-- MSC2000 codes: 30E10, 42C05, 41A20, 30D30.MR#: MR2016676 (2004k:30087)Zbl#: Zbl 1051.30033Using a convergence theorem for Fourier–Padé approximants constructed from orthogonal polynomials on the unit circle, we prove an analogue of Hadamard's theorem for determining the radius of m-meromorphy of a function analytic on the unit disk and apply this to the location of poles of the reciprocal of Szegö functions.The research of D.B.R. and G.L.L. was supported, in part, by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under grant BFM 2000-0206-C04-01 and the research of G.L.L. was also supported by Ministerio da Ciencia e do Ensino Superior, under Grant PRAXIS XXI BCC-22201/99, and by INTAS under Grant 2000-272. The research of E.B.S. was supported, in part, by V.S. National Science Foundation Grant DMS-0296026.Publicad

Asymptotics of orthogonal polynomials inside the unit circle and Szegö-Padé approximants

11 pages, no figures.-- MSC2000 codes: 42C05, 41A21.MR#: MR1858277 (2002h:42043)Zbl#: Zbl 1009.42016We study the asymptotic behavior of orthogonal polynomials inside the unit circle for a subclass of measures that satisfy Szegö's condition. We give a connection between such behavior and a Montessus de Ballore-type theorem for Szegö–Padé rational approximants of the corresponding Szegö function.The research of the second author (G.L.L.) was partially supported by Dirección General de Enseñanza Superior under Grant PB 96-0120-C03-01. The research of the third author (E.B.S.) was supported, partly by NSF research Grant DMS-9801677.Publicad

Approximation of transfer functions of infinite dimensional dynamical systems by rational interpolants with prescribed poles

22 pages, 3 figures.-- MSC2000 codes: 41A21, 41A25.MR#: MR1746794 (2001b:41019)Zbl#: Zbl 0967.41008Rational interpolants with prescribed poles are used to approximate holomorphic functions on the closure of their region of analyticity under natural assumptions of their properties on the boundary. The transfer functions of some infinite dimensional dynamical systems of interest in applications satisfy the restrictions we impose. This is the case for discrete-time fractional filters, time-delay systems, and heat transfer control systems. We give two general results by which, in particular, the transfer functions that arise in such dynamical systems may be approximated. Estimates for the rate of convergence are given. We also include some numerical examples which compare the performance of the method we propose with others commonly used in systems theory.The research by first author (A.R.) was carried under a grant from the German Academic Exchange Service (DAAD). The research by second author (G.L.L.) was partially carried out while the author was visiting Institut für Dynamische Systeme, University of Bremen, for which he is grateful.Publicad

Zero location and n-th root asymptotics of Sobolev orthogonal polynomials

14 pages, no figures.-- MSC2000 codes: 42C05, 33C45.MR#: MR1696589 (2000g:42035)Zbl#: Zbl 0949.42020For a wide class of Sobolev orthogonal polynomials, it is proved that their zeros are contained in a compact subset of the complex plane and the asymptotic zero distribution is obtained. With this information, the nth root asymptotic behavior outside the compact set containing all the zeros is given.Research by the first author (G.L.L.) was partially supported by Dirección General de Enseñanza Superior under Grant PB 96-0120-C03-01 and by INTAS under Grant 93-0219 EXT. Research by the second author (H.P.) carried out while following a Doctoral Program at Universidad Carlos III de Madrid under grant from Agencia Española de Cooperación Internacional.Publicad