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Mat. Sb., 2010, Volume 201, Number 2, Pages 29–78 (Mi msb7515)  

This article is cited in 51 scientific papers (total in 52 papers)

Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Padé approximants

A. I. Aptekarev, V. G. Lysov

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences

Abstract: The paper considers Hermite-Padé approximants to systems of Markov functions defined by means of directed graphs. The minimization problem for the energy functional is investigated for a vector measure whose components are related by a given interaction matrix and supported in some fixed system of intervals. The weak asymptotics of the approximants are obtained in terms of the solution of this problem. The defining graph is allowed to contain undirected cycles, so the minimization problem in question is considered within the class of measures whose masses are not fixed, but allowed to ‘flow’ between intervals. Strong asymptotic formulae are also obtained. The basic tool that is used is an algebraic Riemann surface defined by means of the supports of the components of the extremal measure. The strong asymptotic formulae involve standard functions on this Riemann surface and solutions of some boundary value problems on it. The proof depends upon an asymptotic solution of the corresponding matrix Riemann-Hilbert problem.
Bibliography: 40 titles.

Keywords: Hermite-Padé approximants, multiple orthogonal polynomials, weak and strong asymptotics, extremal equilibrium problems for a system of measures, matrix Riemann-Hilbert problem.
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English version:
Sbornik: Mathematics, 2010, 201:2, 183–234

Bibliographic databases:

UDC: 517.53
MSC: Primary 42C05, 41A21; Secondary 30E25
Received: 22.12.2008 and 03.09.2009

Citation: A. I. Aptekarev, V. G. Lysov, “Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Padé approximants”, Mat. Sb., 201:2 (2010), 29–78; Sb. Math., 201:2 (2010), 183–234

Citation in format AMSBIB
\by A.~I.~Aptekarev, V.~G.~Lysov
\paper Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Pad\'e approximants
\jour Mat. Sb.
\yr 2010
\vol 201
\issue 2
\pages 29--78
\jour Sb. Math.
\yr 2010
\vol 201
\issue 2
\pages 183--234

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    2. Starovoitov A.P., Ryabchenko N.V., Astafeva A.V., “Ob asimptotike sovmestnykh approksimatsii pade dlya dvukh eksponent”, Vesnik Vitsebskaga dzyarzhainaga universiteta, 4:64 (2011), 5–9  elib
    3. E. A. Rakhmanov, “The asymptotics of Hermite-Padé polynomials for two Markov-type functions”, Sb. Math., 202:1 (2011), 127–134  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. A. I. Aptekarev, A. Kuijlaars, “Hermite–Padé approximations and multiple orthogonal polynomial ensembles”, Russian Math. Surveys, 66:6 (2011), 1133–1199  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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