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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