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On equilibrium problems related to the distribution of zeros of the Hermite-Padé polynomials

  • V. I. BuslaevEmail author
  • S. P. Suetin
Article

Abstract

We study two potential-theory equilibrium problems that arise naturally in the theory of the limit distribution of zeros of the Hermite–Padé polynomials. We analyze the relationship between these problems and prove that the equilibrium measure for one of the problems is the balayage of the equilibrium measure for the other problem.

Keywords

STEKLOV Institute Branch Point Equilibrium Problem Rational Approximation Limit Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. I. Aptekarev and D. N. Tulyakov, “The leading term of the Plancherel–Rotach asymptotic formula for solutions of recurrence relations,” Mat. Sb. 205 (12), 17–40 (2014)MathSciNetCrossRefGoogle Scholar
  2. 1.
    A. I. Aptekarev Sb. Math. 205, 1696–1719 (2014)].CrossRefzbMATHGoogle Scholar
  3. 2.
    V. I. Buslaev, “Convergence of multipoint Padé approximants of piecewise analytic functions,” Mat. Sb. 204 (2), 39–72 (2013)CrossRefGoogle Scholar
  4. 2.
    V. I. Buslaev, Sb. Math. 204, 190–222 (2013)].MathSciNetCrossRefGoogle Scholar
  5. 3.
    V. I. Buslaev, A. Martínez-Finkelshtein, and S. P. Suetin, “Method of interior variations and existence of S-compact sets,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 279, 31–58 (2012)Google Scholar
  6. 3.
    V. I. Buslaev, Proc. Steklov Inst. Math. 279, 25–51 (2012)].CrossRefzbMATHGoogle Scholar
  7. 4.
    V. I. Buslaev and S. P. Suetin, “Existence of compact sets with minimum capacity in problems of rational approximation of multivalued analytic functions,” Usp. Mat. Nauk 69 (1), 169–170 (2014)Google Scholar
  8. 4.
    V. I. Buslaev Russ. Math. Surv. 69, 159–161 (2014)].CrossRefzbMATHGoogle Scholar
  9. 5.
    V. I. Buslaev and S. P. Suetin, “On the existence of compacta of minimal capacity in the problems of rational approximation of multi-valued analytic functions,” J. Approx. Theory, doi: 10.1016/j.jat.2015.08.002 (2015); arXiv: 1505.06120 [math.CV].Google Scholar
  10. 6.
    S. Delvaux, A. López, and G. L. López, “A family of Nikishin systems with periodic recurrence coefficients,” Mat. Sb. 204 (1), 47–78 (2013)MathSciNetCrossRefGoogle Scholar
  11. 6.
    S. Delvaux, Sb. Math. 204, 43–74 (2013)].Google Scholar
  12. 7.
    A. A. Gonchar, “Rational approximation of analytic functions,” Sovrem. Probl. Mat. 1, 83–106 (2003)CrossRefGoogle Scholar
  13. 7.
    A. A. Gonchar, Proc. Steklov Inst. Math. 272 (Suppl. 2), S44–S57 (2011)].zbMATHGoogle Scholar
  14. 8.
    A. A. Gonchar and E. A. Rakhmanov, “On convergence of simultaneous Padé approximants for systems of functions of Markov type,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 157, 31–48 (1981)zbMATHGoogle Scholar
  15. 8.
    A. A. Gonchar Proc. Steklov Inst. Math. 157, 31–50 (1983)].zbMATHGoogle Scholar
  16. 9.
    A. A. Gonchar and E. A. Rakhmanov, “Equilibrium measure and the distribution of zeros of extremal polynomials,” Mat. Sb. 125 (1), 117–127 (1984)MathSciNetGoogle Scholar
  17. 9.
    A. A. Gonchar and E. A. Rakhmanov, Math. USSR, Sb. 53, 119–130 (1986)].CrossRefzbMATHGoogle Scholar
  18. 10.
    A. A. Gonchar and E. A. Rakhmanov, “On the equilibrium problem for vector potentials,” Usp. Mat. Nauk 40 (4), 155–156 (1985)Google Scholar
  19. 10.
    A. A. Gonchar and E. A. Rakhmanov, Russ. Math. Surv. 40 (4), 183–184 (1985)].CrossRefzbMATHGoogle Scholar
  20. 11.
    A. A. Gonchar and E. A. Rakhmanov, “Equilibrium distributions and degree of rational approximation of analytic functions,” Mat. Sb. 134 (3), 306–352 (1987)Google Scholar
  21. 11.
    A. A. Gonchar and E. A. Rakhmanov, Math. USSR, Sb. 62, 305–348 (1989)].MathSciNetCrossRefzbMATHGoogle Scholar
  22. 12.
    A. A. Gonchar, E. A. Rakhmanov, and S. P. Suetin, “Padé–Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the S-property of stationary compact sets,” Usp. Mat. Nauk 66 (6), 3–36 (2011)CrossRefGoogle Scholar
  23. 12.
    A. A. Gonchar, Russ. Math. Surv. 66, 1015–1048 (2011)].CrossRefzbMATHGoogle Scholar
  24. 13.
    N. R. Ikonomov, R. K. Kovacheva, and S. P. Suetin, “Some numerical results on the behavior of zeros of the Hermite–Padé polynomials,” arXiv:1501.07090 [math.CV].Google Scholar
  25. 14.
    A. V. Komlov and S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight. II,” Mat. Sb. 205 (9), 121–144 (2014)MathSciNetCrossRefGoogle Scholar
  26. 14.
    A. V. Komlov and S. P. Suetin, Sb. Math. 205, 1334–1356 (2014)].MathSciNetCrossRefzbMATHGoogle Scholar
  27. 15.
    N. S. Landkof, Foundations of Modern Potential Theory (Nauka, Moscow, 1966; Springer, Berlin, 1972).zbMATHGoogle Scholar
  28. 16.
    A. Martínez-Finkelshtein, E. A. Rakhmanov, and S. P. Suetin, “Variation of the equilibrium energy and the S-property of stationary compact sets,” Mat. Sb. 202 (12), 113–136 (2011)CrossRefGoogle Scholar
  29. 16.
    A. Martínez-Finkelshtein, Sb. Math. 202, 1831–1852 (2011)].MathSciNetCrossRefzbMATHGoogle Scholar
  30. 17.
    E. M. Nikishin, “The asymptotic behavior of linear forms for joint Padé approximations,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 33–41 (1986)Google Scholar
  31. 17.
    E. M. Nikishin, Sov. Math. 30 (2), 43–52 (1986)].zbMATHGoogle Scholar
  32. 18.
    E. A. Rakhmanov, “Orthogonal polynomials and S-curves,” in Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications (Am. Math. Soc., Providence, RI, 2012)Google Scholar
  33. 18.
    E. A. Rakhmanov, Contemp. Math. 578, pp. 195–239.Google Scholar
  34. 19.
    E. A. Rakhmanov and S. P. Suetin, “Asymptotic behaviour of the Hermite–Padé polynomials of the 1st kind for a pair of functions forming a Nikishin system,” Usp. Mat. Nauk 67 (5), 177–178 (2012)CrossRefGoogle Scholar
  35. 19.
    E. A. Rakhmanov and S. P. Suetin, Russ. Math. Surv. 67, 954–956 (2012)].CrossRefzbMATHGoogle Scholar
  36. 20.
    E. A. Rakhmanov and S. P. Suetin, “The distribution of the zeros of the Hermite–Padé polynomials for a pair of functions forming a Nikishin system,” Mat. Sb. 204 (9), 115–160 (2013)MathSciNetCrossRefGoogle Scholar
  37. 20.
    E. A. Rakhmanov and S. P. Suetin, Sb. Math. 204, 1347–1390 (2013)].MathSciNetCrossRefzbMATHGoogle Scholar
  38. 21.
    E. B. Saff and V. Totik, Logarithmic Potentials with External Fields (Springer, Berlin, 1997), Grundl. Math. Wiss. 316.Google Scholar
  39. 22.
    H. Stahl, “Asymptotics of Hermite–Padé polynomials and related convergence results: A summary of results,” in Nonlinear Numerical Methods and Rational Approximation (D. Reidel, Dordrecht, 1988)Google Scholar
  40. 22.
    H. Stahl, Math. Appl. 43, pp. 23–53.Google Scholar
  41. 23.
    H. Stahl, “The convergence of Padé approximants to functions with branch points,” J. Approx. Theory 91 (2), 139–204 (1997).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

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

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