On Baker’s Patchwork Conjecture for Diagonal Padé Approximants

Abstract

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.

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Acknowledgements

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].

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Correspondence to D. S. Lubinsky.

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

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Research supported by NSF Grant DMS1800251.

Communicated by Edward B. Saff.

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

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Keywords

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

Mathematics Subject Classification

  • 41A21
  • 41A20
  • 30E10