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On simul taneous rational interpolants of type (α,β)

Approximation Theory

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1354)

Abstract

In this paper a convergence theorem for simultaneous rational interpolants of type (α,β) is established. This is a natural extension of the theorem of Montessus de Ballore for a row sequence of (scalar) Padé approximants. Results on differences of interpolants that tend to zero on some "large region", were first obtained by Walsh when he solved the case of polynomials interpolating on the roots of unity and at the origin. In the second part of this paper we also extend this study to differences of simultaneous rational interpolants of type (α,β).

Keywords

  • Harmonic Function
  • Compact Subset
  • Convergence Theorem
  • Unique Polynomial
  • Pade Approximants

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References

  1. J.L. Walsh.: Interpolation and Approximation by Rational Functions in the Complex Domain. A.M.S. Colloq. Publ. Vol.XX. Providence.R.I. 3-rd Ed 1960.

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  3. Graves Morris and E.B. Saff.: A. de Montessus theorem for vector valued rational interpolants. Rational Approximation and inter polation. Lecture Notes in Math. proceedings, Tampa. Florida. 1983.

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  4. Piedra René.: On differences of approximants type (α,β). Rev. Ciencias Matemáticas, Vol. 6. No.3. 1985 (in spanish).

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  5. Piedra René.: On the differences of simultaneous rational interpo lants. Proceeding of XX Seminar on Banach Center. Poland, (1986). to appear.

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  6. E.R. Saff. A Sharma and R.S. Varqa.: An extension to rational functions of a theorem of J.L.Walsh on differences of interpolating polynomials. R.A.I.R.O. Analyse Numerique/Numerical Analysis. Vol. 15. No-4, 1981.

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© 1988 Springer-Verlag

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Piedra, R. (1988). On simul taneous rational interpolants of type (α,β). In: Gómez-Fernandez, J.A., Guerra-Vázquez, F., López-Lagomasino, G., Jiménez-Pozo, M.A. (eds) Approximation and Optimization. Lecture Notes in Mathematics, vol 1354. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089596

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  • DOI: https://doi.org/10.1007/BFb0089596

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50443-6

  • Online ISBN: 978-3-540-46005-3

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