On Baker’s Patchwork Conjecture for Diagonal Padé Approximants


We prove that for entire functions f of finite order, there is a sequence of integers \(\mathcal {S}\) such that as \(n\rightarrow \infty \) through S,

$$\begin{aligned} \min \left\{ \left| f-\left[ n/n\right] \right| \left( z\right) ,\left| f-\left[ n-1/n-1\right] \right| \left( z\right) \right\} ^{1/n}\rightarrow 0 \end{aligned}$$

uniformly for z in compact subsets of the plane. More generally this holds for sequences of Newton–Padé approximants and for functions whose errors of approximation by rational functions of type \(\left( n,n\right) \) decay sufficiently fast. This establishes George Baker’s patchwork conjecture for large classes of entire functions.

This is a preview of subscription content, access via your institution.


  1. 1.

    Baker, G.A.: Some structural properties of two counter-examples to the Baker–Gammel–Wills conjecture. J. Comput. Appl. Math. 161, 371–391 (2003)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Baker, G.A.: Counter-examples to the Baker–Gammel–Wills conjecture and patchwork convergence. J. Comput. Appl. Math. 179, 1–14 (2005)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Baker, G.A., Gammel, J.L., Wills, J.G.: An investigation of the applicability of the Padé approximant method. J. Math. Anal. Appl. 2, 405–418 (1961)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Baker, G.A., Graves-Morris, P.: Pad é Approximants, 2nd edn. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  5. 5.

    Buslaev, V.I.: The Baker–Gammel–Wills conjecture in the theory of Padé approximants. Math. USSR Sbornik 193, 811–823 (2002)

    Article  Google Scholar 

  6. 6.

    Buslaev, V.I.: Convergence of the Rogers–Ramanujan continued fraction. Math. USSR Sbornik 194, 833–856 (2003)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Caratheodory, C.: Theory of Functions, vol. II. Chelsea, New York (1954)

    MATH  Google Scholar 

  8. 8.

    Gonchar, A.: A local condition of single-valuedness of analytic functions. Math. USSR Sbornik 18, 151–167 (1972)

    Article  Google Scholar 

  9. 9.

    Gonchar, A., Grigorjan, L.: On estimates of the norm of the holomorphic component of a meromorphic function. Math. USSR Sbornik 28, 571–575 (1976)

    Article  Google Scholar 

  10. 10.

    Grigorjan, L.: Estimates of the norm of the holomorphic components of functions meromorphic in domains with a smooth boundary. Math. USSR Sbornik 29, 139–146 (1976)

    Article  Google Scholar 

  11. 11.

    Khristoforov, D.V.: On uniform approximation of elliptic functions by diagonal Padé approximants. Math. Sbornik 200, 923–941 (2009)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lubinsky, D.S.: Padé tables of a class of entire functions. Proc. Am. Math. Soc. 94, 399–405 (1985)

    MATH  Google Scholar 

  13. 13.

    Lubinsky, D.S.: Rogers–Ramanujan and the Baker–Gammel–Wills (Padé) conjecture. Ann. Math. 157, 847–889 (2003)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Lubinsky, D.S.: On uniform convergence of diagonal multipoint Padé approximants for entire functions. Constr. Approx. 49, 149–174 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Nuttall, J.: Convergence of Padé approximants of meromorphic functions. J. Math. Anal. Appl. 31, 147–153 (1970)

    MathSciNet  Article  Google Scholar 

  16. 16.

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

    MathSciNet  Article  Google Scholar 

  17. 17.

    Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  18. 18.

    Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Springer, New York (1997)

    Book  Google Scholar 

Download references


The author thanks the referees for finding many misprints in the original version, as well as for suggestions that improved the presentation, and the reference [11].

Author information



Corresponding author

Correspondence to D. S. Lubinsky.

Additional information

In memory of George A. Baker. Jr., November 25, 1932- July 24, 2018.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research supported by NSF Grant DMS1800251.

Communicated by Edward B. Saff.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lubinsky, D.S. On Baker’s Patchwork Conjecture for Diagonal Padé Approximants. Constr Approx (2021). https://doi.org/10.1007/s00365-020-09525-y

Download citation


  • Padé approximation
  • Multipoint Padé approximants
  • Spurious poles
  • Baker patchwork conjecture

Mathematics Subject Classification

  • 41A21
  • 41A20
  • 30E10