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We construct the FrobeniusPadé approximants of the function |x| in (1,1) for the Chebyshev weight. These rational functions are linked with the FrobeniusPadé approximants of the function x in (0,1). We prove geometric convergence of the approximants of |x| in C{z:(z)=0}. © 2016 Elsevier Inc.

We obtain results on the convergence of Padé approximants of Stieltjes-type meromorphic functions and the relative asymptotics of orthogonal polynomials on unbounded intervals. These theorems extend some results given by Guillermo López in this direction substituting the Carleman condition in his theorems by the determination of the corresponding moment problem. © 2009 Elsevier Inc. All rights reserved.

For any pair of compact intervals of the real line Δ 1 , Δ 2 , with Δ 1 ∩ Δ 2 = ∅ , we obtain two probability measures μ 1 , τ 1 , supported on Δ 1 and Δ 2 respectively, such that the Nikishin system N ( μ 1 , τ 1 ) has a sequence of monic multiple orthogonal polynomials which satisfy a four term recurrence relation with constant coefficients of period 2. The measures are obtained from the functions which give the ratio asymptotic of multiple orthogonal polynomials with respect to an arbitrary Nikishin system N ( σ 1 , σ 2 ) on Δ 1 , Δ 2 , such that σ i ′ > 0 a.e. on Δ i , i = 1 , 2 . The role of μ 1 , τ 1 is symmetric in the sense that the same construction is possible on Δ 2 , Δ 1 , with N ( τ 1 , μ 1 ) .

We study some properties of the zeros and the asymptotic behavior of orthogonal polynomials with respect to varying measures on the unit circle. In the proofs, some techniques of rational approximation are used. © 2005 Elsevier Inc. All rights reserved.

17 pages, no figures.-- MSC1991 codes: Primary: 42C05, 41A20, 65D32; Secondary: 30E10. MR#: MR2036643 (2004m:33021) Zbl#: Zbl 1056.42017 Generalized Stieltjes polynomials are introduced and their asymptotic properties outside the support of the measure are studied. As applications, we prove the convergence of sequences of interpolating rational functions, whose poles are partially fixed, to Markov functions and give an asymptotic estimate of the error of rational Gauss–Kronrod quadrature formulas when functions which are analytic on some neighborhood of the set of integration are considered. The work of M. Bello Hernández and B. de la Calle Ysern was partially supported by Dirección General de Enseñanza Superior (DGES) under grants BFM2000-0206-C04-03 and BFM2000-0206-C04-01, respectively. The work of G. López Lagomasino was partially supported by DGES under grant BFM2000-0206-C04-01 and by INTAS under grant 00-272. Publicado

23 pages, no figures.-- MSC1991 codes: 42C05, 30E10. MR#: MR1702795 (2000j:42036) Zbl#: Zbl 1058.42503 It is known that the common denominator of the Hermite-Padé approximants of a mixed Angelesco-Nikishin system shares orthogonality relations with respect to each function in the system. It is less well known that they also satisfy full orthogonality with respect to a varying measure. This problem motivates our interest in extending the class of varying measures with respect to which weak asymptotics of orthogonal polynomials takes place. In particular, for the case of a Nikishin system, we prove weak asymptotics of the corresponding varying measures. Research by second author (G.L.L.) was supported by Dirección General de Enseñanza Superior under grant PB 96-0120-C03-01 and by INTAS under grant 93-0219 EXT. Publicado

16 pages.-- MSC2000 codes: Primary 42C05, 33C25; Secondary 41A21.-- Submitted paper. For any pair of compact intervals Δ1 and Δ2 of the real line such that Δ1∩Δ2 = ø we obtain two pairs of absolutely continuous probability measures (μ1,μ2) and (τ1,τ2) supported on Δ1 and Δ2, respectively, such that: - for appropriate constants C1 and C2, (μ1,μ2) is the Nikishin system generated by (μ1,C1τ1) and (τ1,τ2) the Nikishin system generated by (τ1,C2μ1), - the polynomials of multiple orthogonality with respect to the Nikishin system (μ1,μ2) and indices {..., (n,n), (n+1,n), ...} satisfy a recurrence relations with constant coe±cients of period 2, - 1/hat-μ1(z) and 1/hat-μ2(z) are the functions which describe the ratio asymptotics of multiple orthogonal polynomials with respect to an arbitrary Nikishin system N(σ1,σ2) verifying supp(σi) = Δi, and σi' > 0, i = 1,2, almost everywhere on Δi. Analogously, 1/hat-τ1(z) and 1/hat-τ2(z) give the ratio asymptotics for N(σ1,σ2). The research of both authors was supported by research grant MTM 2006-13000-C03-02 of Ministerio de Ciencia e Innovación, Spain. I.A. Rocha also received support from Universidad Politécnica de Madrid through Grupo de Investigación TACA. No publicado

26 pages, no figures.-- MSC2000 codes: Primary 42C05, 30E10; Secondary 41A21. MR#: MR2419380 (2009e:42053) Zbl#: Zbl 1153.42013 We 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. Publicado

19 pages, no figures.-- MSC2000 codes: Primary 42C05, 41A21.-- Originally published in Russian language by the Russian Academy of Mathematics in: Matematicheskii Sbornik 196(8): 3–20 (2005). Zbl#: Zbl 1077.42015 The existence of ratio asymptotics is proved for a sequence of multiple orthogonal polynomials with orthogonality relations distributed among a system of m finite Borel measures with support on a bounded interval of the real line which form a so-called Nikishin system. For m=1 this result reduces to Rakhmanov's celebrated theorem on the ratio asymptotics for orthogonal polynomials on the real line. The research of the first author was carried out with the support of the Russian Foundation for Basic Research (grant no. 02-01-00564), the Programme of Support of Leading Scientific Schools of RF (grant no. NSh-1551.2003.1), the RAS Programme ‘Current Problems of Theoretical Mathematics’, INTAS (grant no. 03-516637) and NATO PST.CLG (grant no. 979738). The research of the second author was carried out with the support of INTAS (grant no. 03-516637), NATO PST.CLG (grant no. 979738), and BFM (grant no. 2003-06335-C03-02). The research of the third author was carried out with the support BFM (grant no. 2003-06335-C03-02). Publicado

12 pages, no figures.-- MSC2000 codes: Primary: 42C05, 30E15; Secondary: 30E10. We consider the relative asymptotic behaviour of orthogonal polynomials with respect to varying measures supported on the real line and the unit circle. The main feature of the results is the generality of the class of measures studied. El trabajo de los dos autores ha sido parcialmente financiado por la Dirección General de Investigación, Ministerio de Ciencia y Tecnología, a través del proyecto BFM2000-0206-C04-01 y además, en el caso de G. López, por el INTAS. Publicado

13 pages, no figures.-- MSC1991 codes: 42C05; 30E10.-- Issue title: "Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Application" (Seville, Spain, Sep. 1997), edited by Antonio J. Durán. MR#: MR1662686 (99i:42035) Zbl#: Zbl 0933.42017 The strong asymptotic behaviour of orthogonal polynomials with respect to a general class of varying measures is given for the case of the unit circle and the real line. These results are used to obtain certain asymptotic relations for the polynomials involved in the construction of Hermite-Padé approximants of a Nikishin system of functions. 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 and by INTAS under grant 93-0219 EXT. Publicado

In this paper, we give strong asymptotics of extremal polynomials with respect to varying measures of the form d n = d/|Y n| p, where is a positive measure on a closed analytic Jordan curve C, and {Y n} is a sequence of polynomials such that for each n, Y n has exactly degree n and all its zeros ( n, i), i = 1, 2, ..., lie in the exterior of C. Copyright © 2005, Kent State University.