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Mathematical Notes

, Volume 104, Issue 5–6, pp 905–914 | Cite as

On an Example of the Nikishin System

  • S. P. SuetinEmail author
Article
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Abstract

An example of a Markov function f = const + \(\hat \sigma \) such that the three functions f, f2, and f3 constitute a Nikishin systemis given. It is conjectured that there exists aMarkov function f such that, for each n ∈ N, the system of f, f2,..., fn is a Nikishin system.

Keywords

Hermite–Padé polynomials Angelesco system Nikishin system 

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References

  1. 1.
    J. Nuttall, “Asymptotics of diagonal Hermite–Padépolynomials,” J. Approx. Theory 42 (4), 299–386 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    E. M. Nikishin and V. N. Sorokin, Rational Approximation and Orthogonality (Nauka, Moscow, 1988) [in Russian].zbMATHGoogle Scholar
  3. 3.
    W. Van Assche, “Padéand Hermite–Padéapproximation and orthogonality,” Surv. Approx. Theory 2, 61–91 (2006).MathSciNetGoogle Scholar
  4. 4.
    A. A. Gonchar, E. A. Rakhmanov, and V. N. Sorokin, “Hermite-Padéapproximants for systems of Markov-type functions,” Mat. Sb. 188 (5), 33–58 (1997) [Sb.Math. 188 (5), 671–696 (1997)].MathSciNetCrossRefGoogle Scholar
  5. 5.
    U. Fidalgo Prieto, A. López García, G. López Lagomasino, and V. N. Sorokin, “Mixed type multiple orthogonal polynomials for two Nikishin systems,” Constr. Approx. 32 (No. 10), 255–306 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    U. Fidalgo Prieto and G. López Lagomasino, “Nikishin systems are perfect,” Constr. Approx. 34 (3), 297–356 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    E. A. Rakhmanov, “Zero distribution for Angelesco Hermite–Padépolynomials,” UspekhiMat. Nauk 73 (3 (441)), 89–156 (2018) [RussianMath. Surveys 73 (3), 457–518 (2018)].CrossRefGoogle Scholar
  8. 8.
    V. N. Sorokin, “Simultaneous approximation of several linear forms,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 1, 44–47 (1983).MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. I. Aptekarev and V. G. Lysov, “Systems of Markov functions generated by graphs and the asymptotics of their Hermite–Padéapproximants,” Mat. Sb. 201 (2), 29–78 (2010) [Sb.Math. 201 (2), 183–234 (2010)].MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    E. A. Rakhmanov, “The asymptotics of Hermite–Padépolynomials for two Markov-type functions,” Mat. Sb. 202 (1), 133–140 (2011) [Sb.Math. 202 (1), 127–134 (2011)].MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. P. Suetin, Hermite–PadéPolynomials and Analytic Continuation: New Approach and Some Results, arXiv: http://arxiv.org/abs/1806.08735, 2018.Google Scholar
  12. 12.
    A. V. Komlov, R. V. Pal’velev, S. P. Suetin, and E.M. Chirka, “Hermite–Padéapproximants for meromorphic functions on a compact Riemann surface,” UspekhiMat. Nauk 72 (4 (436)), 95–130 (2017) [RussianMath. Surveys 72 (4), 671–706 (2017)].CrossRefGoogle Scholar
  13. 13.
    A. I. Aptekarev, A. I. Bogolyubskii, and M. L. Yattselev, “Convergence of ray sequences of Frobenius–Padéapproximants,” Mat. Sb. 208 (3), 4–27 (2017) [Sb.Math. 208 (3), 313–334 (2017)].MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D. Barrios Rolanía, J. S. Geronimo, and G. López Lagomasino, “High-order recurrence relations, Hermite–Padéapproximation, and Nikishin systems,” Mat. Sb. 209 (3), 102–137 (2018) [Sb. Math. 209 (3), 385–420 (2018)].MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. López-García and E. Miña-Días, “Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials,” Mat. Sb. 209 (7), 139–177 (2018) [Sb. Math. 209 (7), 1051–1088 (2018)].MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    G. López Lagomasino and W. Van Assche, “Riemann–Hilbert analysis for a Nikishin system,” Mat. Sb. 209 (7), 106–138 (2018) [Sb.Math. 209 (7), 1019–1050 (2018)].MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    S. P. Suetin, “On a new approach to the problem of distribution of zeros of Hermite–Padépolynomials for a Nikishin system,” in Trudy Mat. Inst. Steklova, Vol. 301: Complex Analysis, Mathematical Physics, and Applications, Collected papers, (MAIK Nauka/Interperiodica, Moscow, 2018), pp. 259–275 [Proc. Steklov Inst.Math. 301, 245–261 (2018)].Google Scholar
  18. 18.
    S. P. Suetin, “Distribution of the zeros of Hermite–Padépolynomials for a complex Nikishin system,” for the complex Nikishin system,” Uspekhi Mat. Nauk 73 (2 (440)), 183–184 (2018) [Russian Math. Surveys 73 (2), 363–365 (2018)].CrossRefGoogle Scholar
  19. 19.
    S. P. Suetin, “An analog of Pólya’s theoremformultivalued analytic functions with finitelymany branch points,” Mat. Zametki 101 (5), 779–791 (2017) [Math. Notes 101 (5), 888–898 (2017)].MathSciNetCrossRefGoogle Scholar
  20. 20.
    S. P. Suetin, “On the distribution of the zeros of the Hermite–Padépolynomials for a quadruple of functions,” UspekhiMat. Nauk 72 (2 (434)), 191–192 (2017) [Russian Math. Surveys 72 (2), 375–377 (2017)].CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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