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On Padé approximants of Markov-type meromorphic functions

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Abstract

The paper is devoted to the asymptotic properties of diagonal Padé approximants for Markov-type meromorphic functions. The main result is strong asymptotic formulas for the denominators of diagonal Padé approximants for Markov-type meromorphic functions f = \( \hat \sigma \) + r under additional constraints on the measure σ (r is a rational function). On the basis of these formulas, it is proved that, in a sufficiently small neighborhood of a pole of multiplicity m of such a meromorphic function f, all poles of the diagonal Padé approximants f n are simple and asymptotically located at the vertices of a regular m-gon.

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References

  1. A. Baker, Jr. and P. Graves-Morris, Padé Approximants (Addison-Wesley, Reading, Mass., 1981; Mir, Moscow, 1986).

    Google Scholar 

  2. P. L. Chebyshev, “On Continued Fractions,” Uchen. Zap. Imp. Akad. Sci. III, 636–664 (1855).

    Google Scholar 

  3. A. A. Markov, “Deux dé monstrations de la convergence de certaines fractions continues,” Acta Math. 19, 93–104 (1895).

    Article  MathSciNet  Google Scholar 

  4. A. A. Gonchar, “On the Convergence of Padé Approximants for Some Classes of Meromorphic Functions,” Mat. Sb. 97, 607–629 (1975).

    MathSciNet  Google Scholar 

  5. E. A. Rakhmanov, “On Ratio Asymptotics for Orthogonal Polynomials,” Mat. Sb. 103(2), 237–252 (1977).

    MathSciNet  Google Scholar 

  6. E. A. Rakhmanov, “On Ratio Asymptotics for Orthogonal Polynomials. II,” Mat. Sb. 118, 104–117 (1982).

    MathSciNet  Google Scholar 

  7. A. A. Gonchar, “On the Uniform Convergence of Diagonal Padé Approximants,” Mat. Sb. 118, 535–556 (1982).

    MathSciNet  Google Scholar 

  8. A. A. Gonchar, “On the Convergence of Diagonal Padé Approximants in the Spherical Metric,” in Mathematical Structures-Computational Mathematics-Mathematical Modeling (Sofia, 1984), Vol. 2, pp. 29–35 [in Russian].

  9. S. N. Bernshtein, On Polynomials Orthogonal in a Finite Interval(ONTI, Kharkov, 1937) [in Russian].

    Google Scholar 

  10. N. I. Akhiezer, “On Orthogonal Polynomials on Several Intervals,” Dokl. Akad. Nauk SSSR 134(1), 9–12 (1960).

    MathSciNet  Google Scholar 

  11. N. I. Akhiezer and Yu. Ya. Tomchuk, “To the Theory of Orthogonal Polynomials on Several Intervals,” Dokl. Akad. Nauk SSSR 138(4), 743–746 (1961).

    MathSciNet  Google Scholar 

  12. N. I. Akhiezer, “Continual Analogues of Orthogonal Polynomials on a System of Intervals,” Dokl. Akad. Nauk SSSR 141(2), 263–266 (1961).

    MathSciNet  Google Scholar 

  13. J. Nuttall, “Asymptotics of Diagonal Hermite-Padé Polynomials,” J. Approx. Theory 42, 299–386 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Nuttall, “Padé Polynomial Asymptotics from a Singular Integral Equation,” Constr. Approx. 6(2), 157–166 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  15. S. P. Suetin, “On the Uniform Convergence of Diagonal Padé Approximants for Hyperelliptic Functions,” Mat. Sb. 191(9), 81–114 (2000) [Russ. Acad. Sci. Sb. Math. 191 (9), 1339–1373 (2000)].

    MathSciNet  Google Scholar 

  16. S. P. Suetin, “Padé Approximants and Efficient Analytic Continuation of a Power Series,” Usp. Mat. Nauk 57(1), 45–142 (2002) [Russ. Math. Surveys 57 (1), 43–141 (2002)].

    MathSciNet  Google Scholar 

  17. S. P. Suetin, “Approximation Properties of the Poles of Diagonal Padé Approximants for Certain Generalizations of Markov Functions,” Mat. Sb. 193(12), 81–114 (2002) [Russ. Acad. Sci. Mat. Sb. 193 (12), 1837–1866 (2002)].

    MathSciNet  Google Scholar 

  18. A. I. Aptekarev, “Sharp Constants for Rational Approximants Analytic Functions,” Mat. Sb. 193(1), 3–72 (2002) [Mat. Sb. 193 (1), 1–72 (2002)].

    MathSciNet  Google Scholar 

  19. E. I. Zverovich, “Boundary Value Problems of the Theory of Analytic Functions in Hölder Classes on Riemann Surfaces,” Usp. Mat. Nauk 26(1), 113–180 (1971).

    MATH  Google Scholar 

  20. E. A. Rakhmanov, “On the Convergence of Diagonal Padé Approximants,” Mat. Sb. 104, 271–291 (1977).

    MathSciNet  Google Scholar 

  21. O. Forster, Lectures on Riemannian Surfaces (Mir, Moscow, 1980; Springer, New York, 1981).

    Google Scholar 

  22. B. A. Dubrovin, “Theta-Functions and Nonlinear Equations,” Usp. Mat. Nauk 36(2), 11–80 (1981).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    A. A. Gonchar & S. P. Suetin

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  1. A. A. Gonchar
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  2. S. P. Suetin
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Original Russian Text © A.A. Gonchar and S.P. Suetin, 2004, published in Sovremennye Problemy Matematiki, 2004, Vol. 5, pp. 3–66.

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Gonchar, A.A., Suetin, S.P. On Padé approximants of Markov-type meromorphic functions. Proc. Steklov Inst. Math. 272 (Suppl 2), 58–95 (2011). https://doi.org/10.1134/S0081543811030059

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