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

This article is cited in 37 scientific papers (total in 38 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.

DOI: https://doi.org/10.4213/sm7515

Full text: PDF file (1028 kB)
References: PDF file   HTML file

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

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. N. Sorokin, “On multiple orthogonal polynomials for discrete Meixner measures”, Sb. Math., 201:10 (2010), 1539–1561  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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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    6. A. I. Aptekarev, V. G. Lysov, D. N. Tulyakov, “Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials”, Sb. Math., 202:2 (2011), 155–206  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of Meixner polynomials and Christoffel-Darboux kernels”, Trans. Moscow Math. Soc., 73 (2012), 67–106  mathnet  crossref  mathscinet  zmath  elib
    8. N. V. Ryabchenko, A. P. Starovoitov, G. N. Kazimirov, “Ermitovskaya approksimatsiya dvukh eksponent”, PFMT, 2012, no. 1(10), 97–100  mathnet
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    11. E. A. Rakhmanov, S. P. Suetin, “Asymptotic behaviour of the Hermite–Padé polynomials of the 1st kind for a pair of functions forming a Nikishin system”, Russian Math. Surveys, 67:5 (2012), 954–956  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. Beckermann B., Kalyagin V., Matos A.C., Wielonsky F., “Equilibrium problems for vector potentials with semidefinite interaction matrices and constrained masses”, Constr. Approx., 37:1 (2013), 101–134  crossref  mathscinet  zmath  isi  elib
    13. A. P. Starovoitov, “Ermitovskaya approksimatsiya dvukh eksponent”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:1(2) (2013), 87–91  mathnet
    14. A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag-Lefflera”, PFMT, 2013, no. 1(14), 81–87  mathnet
    15. Baratchart L., Yattselev M. L., “Padé approximants to certain elliptic-type functions”, J. Anal. Math., 121 (2013), 31–86  crossref  mathscinet  zmath  isi  elib
    16. E. A. Rakhmanov, S. P. Suetin, “The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system”, Sb. Math., 204:9 (2013), 1347–1390  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    17. S. Delvaux, A. López, G. López Lagomasino, “A family of Nikishin systems with periodic recurrence coefficients”, Sb. Math., 204:1 (2013), 43–74  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    18. N. Zorii, “Necessary and sufficient conditions for the solvability of the Gauss variational problem for infinite dimensional vector measures”, Potential Anal., 41:1 (2014), 81–115  crossref  mathscinet  zmath  isi  elib
    19. R. K. Kovacheva, S. P. Suetin, “Distribution of zeros of the Hermite–Padé polynomials for a system of three functions, and the Nuttall condenser”, Proc. Steklov Inst. Math., 284 (2014), 168–191  mathnet  crossref  crossref  isi
    20. A. Aptekarev, J. Arvesú, “Asymptotics for multiple Meixner polynomials”, J. Math. Anal. Appl., 411:2 (2014), 485–505  crossref  mathscinet  isi  elib
    21. M. A. Lapik, “Formula Buyarova–Rakhmanova dlya vneshnego polya v vektornoi zadache ravnovesiya logarifmicheskogo potentsiala”, Preprinty IPM im. M. V. Keldysha, 2014, 082, 15 pp.  mathnet
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    23. A. P. Starovoitov, “On asymptotic form of the Hermite–Pade approximations for a system of Mittag-Leffler functions”, Russian Math. (Iz. VUZ), 58:9 (2014), 49–56  mathnet  crossref
    24. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    25. V. M. Buchstaber, V. N. Dubinin, V. A. Kaliaguine, B. S. Kashin, V. N. Sorokin, S. P. Suetin, D. N. Tulyakov, B. N. Chetverushkin, E. M. Chirka, A. A. Shkalikov, “Alexander Ivanovich Aptekarev (on his 60th birthday)”, Russian Math. Surveys, 70:5 (2015), 965–973  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    26. Aptekarev A.I. Yattselev M.L., “Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials”, Acta Math., 215:2 (2015), 217–280  crossref  mathscinet  zmath  isi  scopus
    27. Aptekarev A.I. Lopez Lagomasino G. Martinez-Finkelshtein A., “Strong asymptotics for the Pollaczek multiple orthogonal polynomials”, Dokl. Math., 92:3 (2015), 709–713  crossref  mathscinet  zmath  isi  scopus
    28. M. A. Lapik, “Families of vector measures which are equilibrium measures in an external field”, Sb. Math., 206:2 (2015), 211–224  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    29. V. N. Sorokin, “Ob asimptoticheskikh rezhimakh sovmestnykh mnogochlenov Meiksnera”, Preprinty IPM im. M. V. Keldysha, 2016, 046, 32 pp.  mathnet  crossref
    30. V. G. Lysov, D. N. Tulyakov, “O vektornoi teoretiko-potentsialnoi zadache s matritsei Anzhelesko”, Preprinty IPM im. M. V. Keldysha, 2016, 110, 36 pp.  mathnet  crossref
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    33. Aptekarev A.I. Van Assche W. Yattselev M.L., “Hermite-Padé Approximants for a Pair of Cauchy Transforms with Overlapping Symmetric Supports”, Commun. Pure Appl. Math., 70:3 (2017), 444–510  crossref  mathscinet  zmath  isi  scopus
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