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Numerische Mathematik

, Volume 73, Issue 3, pp 311–327 | Cite as

Kronecker type theorems, normality and continuity of the multivariate Padé operator

  • Annie Cuyt
  • Kathy Driver
  • Doron Lubinsky

Summary.

For univariate functions the Kronecker theorem, stating the equivalence between the existence of an infinite block in the table of Padé approximants and the approximated function \(f\) being rational, is well-known. In [Lubi88] Lubinsky proved that if \(f\) is not rational, then its Padé table is normal almost everywhere: for an at most countable set of points the Taylor series expansion of \(f\) is such that it generates a non-normal Padé table. This implies that the Padé operator is an almost always continuous operator because it is continuous when computing a normal Padé approximant [Wuyt81]. In this paper we generalize the above results to the case of multivariate Padé approximation. We distinguish between two different approaches for the definition of multivariate Padé approximants: the general order one introduced in [Levi76, CuVe84] and the so-called homogeneous one discussed in [Cuyt84].

Mathematics Subject Classification (1991):65D15 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Annie Cuyt
    • 1
  • Kathy Driver
    • 2
  • Doron Lubinsky
    • 2
  1. 1.NWFO (Belgium), Department Wis-Inf, University of Antwerp (UIA), Universiteitsplein 1, B-2610 Antwerp, Belgium BE
  2. 2.Department Mathematics, Witwatersrand University, P.O. Wits 2050, South Africa ZA

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