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Three different approaches to a proof of convergence for Padé approximants

Polynomial And Rational Approximation

Part of the Lecture Notes in Mathematics book series (LNM,volume 1237)

Abstract

Three different ways of proving the convergence of close-to-diagonal sequences of Padé approximants to functions with branch points are compared. It is assumed that the functions to be approximated have all their singularities in a compact set of capacity zero.

Keywords

  • Riemann Surface
  • Branch Point
  • Orthogonal Polynomial
  • Integration Path
  • Quadratic Differential

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.

Research supported in part by Natural Sciences and Engineering Research Council Canada.

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References

  1. Nuttall, J. (1977). The convergence of Padé approximants to functions with branch points, in "Padé and Rationa Approximation" (E.B. Saff and R.S. Varga, Eds.), Academic Press, New York, pp. 101–109.

    CrossRef  Google Scholar 

  2. Nuttall, J. and Singh, S.R. (1978): Orthogonal polynomials and Padé approximants associated with a system of arcs, J. Approx. Theory, 21, pp. 1–42.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Stahl, H. (1982): The convergence of Padé approximants to functions with branch points, submitted to J. Approx. Theory.

    Google Scholar 

  4. Stahl, H. (1985): The convergence of generalized Padé approximants, will appear in J. Constr. Approx.

    Google Scholar 

  5. Nuttall, J. (1982): The convergence of Padé approximants and their generalizations, Springer Lect. Notes Math., 925, pp. 246–257.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Nuttall, J. (1984): Asymptotics of diagonal Hermite-Padé polynomials, J. Approx. Theory, 42, pp. 299–386.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Nuttall, J. (1985): On sets of minimum capacity, in "Lecture Notes in Pure and Applied Mathematics" Dekker, New York.

    Google Scholar 

  8. Stahl, H. (1985): Extremal domains associated with an analytic function I and II, Complex Variables, 4, pp. 311–324, 325–338.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Nuttall, J. (1980): Sets of minimum capacity, Padé approximants and the bubble problem, in "Bifurcation Phenomena in Mathematical Physics and Related Topics" (C. Bardos and D. Bessis, Eds.), Reidel, Dordrecht, pp. 185–201.

    CrossRef  Google Scholar 

  10. Stahl, H. (1985): On the divergence of certain Padé approximants and the behaviour of the associated orthogonal polynomials, in "Poly-Springer Lect. Notes Math., 1171, pp. 321–330.

    CrossRef  MathSciNet  Google Scholar 

  11. Rahmanov, E.A. (1980): On the convergence of Padé approximants in classes of holomorphic functions, Matem. Sbornik, 112 (154). English translation: Math. USSR Sb., 40 (1981), pp. 149–155.

    Google Scholar 

  12. Stahl, H. (1985): Divergence of diagonal Padé approximants and the asymptotic behavior of orthogonal polynomials associated with non-positive measures, Constr. Approx., 1, pp. 249–270.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Stahl, H. (1984): Convergence in capacity and uniform convergence, will appear in J. Approx. Theory.

    Google Scholar 

  14. Pommerenke, Ch. (1973): Padé approximants and convergence in capacity, J. Math. Anal. Appl., 41, pp. 775–780.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Stahl, H. (1976): Beiträge zum Problem der Konvergenz von Padéapproximierenden, doctorial dissertation, Technical University Berlin.

    Google Scholar 

  16. Graves-Morris, P.R. (1981): The convergence of ray sequences of Padé approximants of Stiltjes functions, J. Comp. Appl. Math., 7, pp. 191–201.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Walsh, J.L. (1964): Padé approximants as limits of rational functions of best approximation, J. Math. Mech., 13, pp. 305–312.

    MathSciNet  MATH  Google Scholar 

  18. Trefethen, L.N. and Gutknecht, M.H. (1985): On convergence and degeneration in rational Padé and Chebyshev approximation, SIAM J. Math. Anal., 16, pp. 198–210.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. Stahl, H. (1985): The structure of extremal domain associated with an analytic function, Complex Variables, 4, pp. 339–354.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Jensen, G. (1975): Quadratic Differentials, Chap. 8 in "Univalent Functions" by Ch. Pommerenke, Vandenhoeck and Ruprecht, Göttingen.

    Google Scholar 

  21. Gammel, J.L. and Nuttall, J. (1982): Note on generalized Jacobi polynomials, Springer Lect. Notes Math., 925, pp. 258–270.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. Nuttall, J. (1985): Asymptotics of generalized Jacobi polynomials, will appear in Constr. Approx.

    Google Scholar 

  23. Szegö, G. (1959): Orthogonal Polynomials, American Mathematical Society, New York.

    MATH  Google Scholar 

  24. Siegel, C.L. (1971): Topics in Complex Function Theory, Interscience, New York.

    MATH  Google Scholar 

  25. Faber, G. (1922): Über nach Polynomen fortschreitende Reihen, Sitzungsberichte der Bayrischen Akademie der Wissenschaften, pp. 157–178.

    Google Scholar 

  26. Landkof, N.S. (1972): Foundations of Modern Potential Theory, Springer Verlag, Berlin.

    CrossRef  MATH  Google Scholar 

  27. Stahl, H. (1985): Orthogonal polynomials with complex valued weight functions I, will appear in Constr. Approx.

    Google Scholar 

  28. Stahl, H. (1985): Orthogonal polynomials with complex valued weight Functions II, will appear in Constr. Approx.

    Google Scholar 

  29. Baxter, G (1961): A convergence equivalence related to polynomials orthogonal on the unit circle., Trans. Amer. Math. Soc., 99, pp. 471–487.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. Nuttall, J. (unpublished manuscript): Orthogonal polynomials for complex weight functions and the convergence of related Padé approximants.

    Google Scholar 

  31. Magnus, A (1986): Toeplitz matrix techniques and convergence of complex weight Padé approximants, will appear in J. Comp. Appl. Math.

    Google Scholar 

  32. Widom, H. (1969): Extremal polynomials associated with a system of curves in the complex plane, Advances in Math., 3, pp. 127–232.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. Grötzsch, C. (1930): Über ein Variationsproblem der konformen Abbildung, Berichte der Sächsischen Akademie der Wissenschaften, Math.-Phys. Kl., 82, pp. 251–263.

    Google Scholar 

  34. Goncar, A.A. and Rahmanov, E.A. (1981): On the convergence of simultaneous Padé approximants for systems of functions of Markov type, Proc. Steklov Math. Inst., 157, pp. 31–48.

    Google Scholar 

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© 1987 Springer-Verlag

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Stahl, H. (1987). Three different approaches to a proof of convergence for Padé approximants. In: Gilewicz, J., Pindor, M., Siemaszko, W. (eds) Rational Approximation and its Applications in Mathematics and Physics. Lecture Notes in Mathematics, vol 1237. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072458

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  • DOI: https://doi.org/10.1007/BFb0072458

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