BIT Numerical Mathematics

, Volume 28, Issue 3, pp 651–658 | Cite as

On the stability of rational approximations to the cosine with only imaginary poles

  • Laurence A. Bales
  • Ohannes A. Karakashian
  • Steve M. Serbin
Part III Numerical Mathematics
  • 13 Downloads

Abstract

Letr(z) be a rational approximation to cosz with only imaginary poles ± 1 −1/2 , ± 2 −1/2 , ..., ± m −1/2 such that |cozzr(z)| ≤C|z|2m+2 as |z| → 0. If the degree of the numerator ofr(z) is less than or equal to 2m andγim/4,i=1, ...,m, then we show that |r(z)|≦1 for all realz.

AMS Subject Classification

65L07 

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References

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    G. A. Baker, V. A. Dougalis and S. M. Serbin,An approximation theorem for second-order evolution equations, Numer. Math. 35 (1980), 127–142.Google Scholar
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    L. A. Bales, O. A. Karakashian and S. M. Serbin,On the A 0-stability of rational approximations to the exponential function with only real poles, BIT 28 (1988), 70–79.Google Scholar
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    S. P. Nørsett and G. Wanner,The real-pole sandwich for rational approximations and oscillation equations, BIT 19 (1979), 79–94.Google Scholar
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    S. M. Serbin,Rational approximation of trigonometric matrices with applications to second-order systems of differential equations, Appl. Math. Comput. 5 (1979), 75–92.Google Scholar

Copyright information

© BIT Foundations 1988

Authors and Affiliations

  • Laurence A. Bales
    • 1
  • Ohannes A. Karakashian
    • 1
  • Steve M. Serbin
    • 1
  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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