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Mat. Sb., 1999, Volume 190, Number 5, Pages 3–44 (Mi msb401)  

This article is cited in 49 scientific papers (total in 50 papers)

Strong asymptotics of multiply orthogonal polynomials for Nikishin systems

A. I. Aptekarev

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

Abstract: Strong asymptotic formulae for the Hermite–Padé polynomials for systems of Markov-type functions in the Nikishin class are obtained. The proof is based on the properties of certain rational functions on Riemann surfaces associated with the supports of the measures generating the Nikishin system.


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English version:
Sbornik: Mathematics, 1999, 190:5, 631–669

Bibliographic databases:

UDC: 517.5
MSC: Primary 42C05, 33C25; Secondary 41A21
Received: 02.02.1998 and 28.09.1998

Citation: A. I. Aptekarev, “Strong asymptotics of multiply orthogonal polynomials for Nikishin systems”, Mat. Sb., 190:5 (1999), 3–44; Sb. Math., 190:5 (1999), 631–669

Citation in format AMSBIB
\by A.~I.~Aptekarev
\paper Strong asymptotics of multiply orthogonal polynomials for Nikishin systems
\jour Mat. Sb.
\yr 1999
\vol 190
\issue 5
\pages 3--44
\jour Sb. Math.
\yr 1999
\vol 190
\issue 5
\pages 631--669

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    This publication is cited in the following articles:
    1. Aptekarev, AI, “Multiple orthogonal polynomials”, Journal of Computational and Applied Mathematics, 99:1–2 (1998), 423  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    2. Van Assche W., Geronimo J.S., Kuijlaars A.B.J., “Riemann–Hilbert problems for multiple orthogonal polynomials”, Special Functions 2000: Current Perspective and Future Directions, Nato Science Series, Series II: Mathematics, Physics and Chemistry, 30, 2001, 23–59  mathscinet  zmath  isi
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    4. A. I. Aptekarev, “Sharp constants for rational approximations of analytic functions”, Sb. Math., 193:1 (2002), 1–72  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. A. I. Aptekarev, G. López Lagomasino, I. Alvarez Rocha, “Ratio asymptotics of Hermite–Padé polynomials for Nikishin systems”, Sb. Math., 196:8 (2005), 1089–1107  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. M. A. Lapik, “Support of the extremal measure in a vector equilibrium problem”, Sb. Math., 197:8 (2006), 1205–1221  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. Aptekarev, AI, “On the limit behavior of recurrence coefficients for multiple orthogonal polynomials”, Journal of Approximation Theory, 139:1–2 (2006), 346  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    8. Enolskii V., Grava T., “Riemann–Hilbert problem and algebraic curves”, Bilinear Integrable Systems: From Classical to Quatum, Continuous to Discrete, Nato Science Series, Series II: Mathematics, Physics and Chemistry, 201, 2006, 65–76  mathscinet  zmath  isi
    9. Aptekarev, AI, “Asymptotics of Hermite-Pade Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0)”, International Mathematics Research Papers, 2008, rpm007  mathscinet  zmath  isi  elib
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    11. Aptekarev, AI, “Asymptotics of Hermite-Pade Approximants for Two Functions with Branch Points”, Doklady Mathematics, 78:2 (2008), 717  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    12. Balogh, F, “Regularity of a vector potential problem and its spectral curve”, Journal of Approximation Theory, 161:1 (2009), 353  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    13. Aptekarev, AI, “Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials”, Constructive Approximation, 30:2 (2009), 175  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    14. A. I. Aptekarev, V. G. Lysov, “Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Padé approximants”, Sb. Math., 201:2 (2010), 183–234  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    15. Lopez Lagomasino G., Rocha I.A., “A canonical family of multiple orthogonal polynomials for Nikishin systems”, Journal of Mathematical Analysis and Applications, 372:2 (2010), 390–401  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    16. de la Calle Ysern B., “A Walk through Approximation Theory”, Recent Trends in Orthogonal Polynomials and Approximation Theory, Contemporary Mathematics, 507, 2010, 25–86  crossref  mathscinet  zmath  isi
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    19. A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    20. 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
    21. Fidalgo Prieto U., Lopez Lagomasino G., “Nikishin Systems Are Perfect”, Constr Approx, 34:3 (2011), 297–356  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    22. Beckermann B., Kalyagin V., Matos A.C., Wielonsky F., “How Well Does the Hermite-Pade Approximation Smooth the Gibbs Phenomenon?”, Math Comp, 80:274 (2011), 931–958  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
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    24. 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
    25. A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag-Lefflera”, PFMT, 2013, no. 1(14), 81–87  mathnet
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    30. A. I. Aptekarev, A. I. Bogolyubskii, “Matrichnaya zadacha Rimana–Gilberta dlya approksimatsii Pade po ortogonalnym razlozheniyam”, Preprinty IPM im. M. V. Keldysha, 2014, 103, 16 pp.  mathnet
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    34. M. A. Lapik, “Ekstremalnyi funktsional dlya vektornykh zadach ravnovesiya logarifmicheskogo potentsiala vo vneshnem pole s matritsei vzaimodeistviya Anzhelesko”, Preprinty IPM im. M. V. Keldysha, 2015, 083, 23 pp.  mathnet
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    39. 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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    41. V. G. Lysov, “Ob approksimatsiyakh Ermita–Pade dlya proizvedeniya dvukh logarifmov”, Preprinty IPM im. M. V. Keldysha, 2017, 141, 24 pp.  mathnet  crossref
    42. Lopez-Garcia A. Lopez Lagomasino G., “Nikishin Systems on Star-Like Sets: Ratio Asymptotics of the Associated Multiple Orthogonal Polynomials”, J. Approx. Theory, 225 (2018), 1–40  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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    44. G. López Lagomasino, W. Van Assche, “Riemann-Hilbert analysis for a Nikishin system”, Sb. Math., 209:7 (2018), 1019–1050  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    45. S. P. Suetin, “On a new approach to the problem of distribution of zeros of Hermite–Padé polynomials for a Nikishin system”, Proc. Steklov Inst. Math., 301 (2018), 245–261  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    46. M. A. Lapik, D. N. Tulyakov, “Vektornye zadachi ravnovesiya Anzhelesko i Nikishina i mnogomernyi analog tsepochki Tody. Predelnyi sluchai, kogda vtoroi otrezok styanut v tochku”, Preprinty IPM im. M. V. Keldysha, 2018, 275, 12 pp.  mathnet  crossref  elib
    47. Lopez-Garcia A. Lopez Lagomasino G., “Nikishin Systems on Star-Like Sets: Ratio Asymptotics of the Associated Multiple Orthogonal Polynomials, II”, J. Approx. Theory, 250 (2020), UNSP 105320  crossref  mathscinet  isi
    48. V. G. Lysov, “Mixed Type Hermite–Padé Approximants for a Nikishin System”, Proc. Steklov Inst. Math., 311 (2020), 199–213  mathnet  crossref  crossref  mathscinet  isi  elib
    49. Bosuwan N., “On Row Sequences of Hermite-Pade Approximation and Its Generalizations”, Mathematics, 8:3 (2020)  crossref  isi
    50. A. I. Aptekarev, “Slabye i silnye asimptotiki ortogonalnykh mnogochlenov c «peremennym» vesom”, Posvyaschaetsya 70-letiyu prezidenta RUDN V.M. Filippova, SMFN, 67, no. 3, Rossiiskii universitet druzhby narodov, M., 2021, 427–441  mathnet  crossref
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