Constructive Approximation

, Volume 3, Issue 1, pp 307–330 | Cite as

Uniform convergence of rows of the Padé table for functions with smooth Maclaurin series coefficients

  • D. S. Lubinsky
Article

Abstract

Given a formal power seriesf(z)≔∑ j=0 ajz j for which the quantitya j −1a j +1/a j 2 has a prescribed asymptotic behavior asj→∞, we obtain the asymptotic behavior of poles of rows of the Padé table, and the associated Toeplitz determinants. In particular, we can show for large classes of entire functions of zero, finite, and infinite order (including the Mittag-Leffler functions) and forn=1,2,3,..., that the poles of [m/n](z) approach ∞ with ratea m /am+1 asm→∞.

Key words and phrases

Padé approximant Toeplitz determinant Asymptotic behavior Uniform convergence Entire functions Padé rows 

AMS classification

Primary 41A21 Secondary 30E05 30E10 

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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • D. S. Lubinsky
    • 1
  1. 1.National Research Institute for Mathematical SciencesCSIRPretoriaRepublic of South Africa

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