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Constructive Approximation

, Volume 49, Issue 1, pp 149–174 | Cite as

On Uniform Convergence of Diagonal Multipoint Padé Approximants for Entire Functions

  • D. S. Lubinsky
Article
  • 162 Downloads

Abstract

We prove that for most entire functions f in the sense of category, a strong form of the Baker–Gammel–Wills conjecture holds. More precisely, there is an infinite sequence \({\mathcal {S}}\) of positive integers n, such that given any \(r>0\), and multipoint Padé approximants \(R_{n}\) to f with interpolation points in \(\left\{ z:\left| z\right| \le r\right\} \), \(\left\{ R_{n}\right\} _{n\in S}\) converges locally uniformly to f in the plane. The sequence \({\mathcal {S}}\) does not depend on r, or on the interpolation points. For entire functions with smooth rapidly decreasing coefficients, full diagonal sequences of multipoint Padé approximants converge.

Keywords

Padé approximation Multipoint Padé approximants Spurious poles 

Mathematics Subject Classification

41A21 41A20 30E10 

References

  1. 1.
    Aptekarev, A.I., Yattselev, M.L.: Padé approximants for functions with branch points—strong asymptotics of Nuttall–Stahl polynomials. Acta Math. 215, 217–280 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baker, G.A., Graves-Morris, P.: Padé Approximants, 2nd edn. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  3. 3.
    Baker, G.A.: Some structural properties of two counter-examples to the Baker–Gammel–Wills conjecture. J. Comput. Appl. Math. 161, 371–391 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baker, G.A.: Counter-examples to the Baker–Gammel–Wills conjecture and patchwork convergence. J. Comput. Appl. Math. 179, 1–14 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beckermann, B., Labahn, G., Matos, A.C.: On rational functions without Froissart doublets. Numer. Math. (to appear)Google Scholar
  6. 6.
    Buslaev, V.I.: The Baker–Gammel–Wills conjecture in the theory of Padé approximants. Math. USSR Sb. 193, 811–823 (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Buslaev, V.I.: Convergence of the Rogers–Ramanujan continued fraction. Math. USSR Sb. 194, 833–856 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buslaev, V.I., Gončar, A.A., Suetin, S.P.: On convergence of subsequences of the mth row of a Padé table. Math. USSR Sb. 48, 535–540 (1984)CrossRefzbMATHGoogle Scholar
  9. 9.
    Cala Rodrigues, F., Lopez Lagomasino, G.: Exact rates of convergence of multipoint Padé approximants. Constr. Approx. 14, 259–272 (1988)CrossRefzbMATHGoogle Scholar
  10. 10.
    Claeys, T., Wielonsky, F.: On sequences of rational interpolants of the exponential function with unbounded interpolation points. J. Approx. Theory 171, 1–32 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Driver, K.A.: Simultaneous rational approximants for a pair of functions with smooth Maclaurin series coefficients. J. Approx. Theory 83, 308–329 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gilewicz, J., Kryakin, Y.: Froissart doublets in Padé approximation in the case of polynomial noise. J. Comput. Appl. Math. 153, 235–242 (2003)Google Scholar
  13. 13.
    Gonchar, A., Lopez Lagomasino, G.: Markov’s theorem for multipoint Padé approximants. Math. Sb. 105, 512–524 (1978)Google Scholar
  14. 14.
    Khristoforov, D.V.: On the phenomenon of spurious interpolation of elliptic functions by diagonal Padé approximants. Math. Notes 87, 564–574 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kovacheva, R.: Zeros of Padé error functions for functions with smooth Maclaurin coefficients. J. Approx. Theory 83, 371–391 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Labych, Y., Starovoitov, A.: Trigonometric Padé approximants for functions with regularly decreasing Fourier coefficients. Math. Sb. 200, 1051–1074 (2009)CrossRefzbMATHGoogle Scholar
  17. 17.
    Lancaster, P., Tismenetsky, M.: The Theory of Matrices. Academic Press, San Diego (1985)zbMATHGoogle Scholar
  18. 18.
    Levin, E., Lubinsky, D.S.: Rows and diagonals of the Walsh array for entire functions with smooth Maclaurin series coefficients. Constr. Approx. 6, 257–286 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lubinsky, D.S.: Padé tables of a class of entire functions. Proc. Am. Math. Soc. 94, 399–405 (1985)zbMATHGoogle Scholar
  20. 20.
    Lubinsky, D.S.: Padé tables of entire functions of very slow and smooth growth. Constr. Approx. 1, 349–358 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lubinsky, D.S.: Padé tables of entire functions of very slow and smooth growth II. Constr. Approx. 4, 321–339 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lubinsky, D.S.: Distribution of poles of diagonal rational approximants to functions of fast rational approximability. Constr. Approx. 7, 501–519 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lubinsky, D.S.: Spurious poles in diagonal rational approximation. In: Gon čar, A.A., Saff, E.B. (eds.) Progress in Approximation Theory, pp. 191–213. Springer, Berlin (1992)CrossRefGoogle Scholar
  24. 24.
    Lubinsky, D.S.: Rogers–Ramanujan and the Baker–Gammel–Wills (Padé) conjecture. Ann. Math. 157, 847–889 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lubinsky, D.S.: Exact interpolation, spurious poles, and uniform convergence of multipoint Padé approximants. Math. Sb. (to appear)Google Scholar
  26. 26.
    Milne-Thomson, L.M.: The Calculus of Finite Differences. Chelsea, New York (1933)zbMATHGoogle Scholar
  27. 27.
    Rakhmanov, E.A.: On the convergence of Padé approximants in classes of holomorphic functions. Math. USSR Sb. 40, 149–155 (1981)CrossRefzbMATHGoogle Scholar
  28. 28.
    Rusak, N., Starovoitov, A.P.: Pad é approximants for entire functions with regularly decreasing Taylor coefficients. Math. Sb. 193(9), 63–92 (2002). [Russian Acad. Sci. Sb. Math. 193 (9), 1303–1332 (2002)]Google Scholar
  29. 29.
    Stahl, H.: Spurious poles in Padé approximation. J. Comput. Appl. Math. 9, 511–527 (1998)CrossRefzbMATHGoogle Scholar
  30. 30.
    Suetin, S.P.: Distribution of the zeros of Pad é polynomials and analytic continuation. Russ. Math. Surv. 70, 901–951 (2015)CrossRefzbMATHGoogle Scholar
  31. 31.
    Wallin, H.: The convergence of Padé approximants and the size of the power series coefficients. Appl. Anal. 4, 235–251 (1974)CrossRefzbMATHGoogle Scholar
  32. 32.
    Wielonsky, F.: Riemann–Hilbert analysis and uniform convergence of rational interpolants to the exponential function. J. Approx. Theory 131, 100–148 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Yattselev, M.: Meromorphic approximation: symmetric contours and wandering poles, manuscriptGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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