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Padé tables of entire functions of very slow and smooth growth

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Letf(z)=σ j−o a j z j be entire with

$$|a_{j - 1} a_{j + 1} /a_j^2 | \leqslant \rho _0^2 ,j = 1,2,3, \ldots ,$$

whereρ 0=0.4559... is the positive root of the equation

$$2\sum\limits_{j = 1}^\infty {\rho ^{j^2 } = 1.}$$


It is shown that the Padé table off is normal, and asL→∞, [L/M L ](z) converges uniformly in compact subsets ofC tof, for any sequence of nonnegative integers {M L } L=1 . In particular, the diagonal sequence {[L/L]} converges uniformly in compact subsets ofC tof. Furthermore, the constantρ 0 is shown to be best possible in a strong sense.

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Communicated by Edward B. Saff.

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Lubinsky, D.S. Padé tables of entire functions of very slow and smooth growth. Constr. Approx 1, 349–358 (1985).

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AMS (MOS) classification

  • Primary
  • 41A21
  • Secondary
  • 30E05
  • 30E10

Key words ana phrases

  • Padé table
  • Uniform convergence
  • Entire functions of slow growth
  • Diagonally
  • dominant matrices