Square blocks and equioscillation in the Padé, walsh, and cf tables

  • Lloyd N. Trefethen
Block Structure
Part of the Lecture Notes in Mathematics book series (LNM, volume 1105)

Abstract

It is well known that degeneracies in the form of repeated entries always occupy square blocks in the Padé table, and likewise in the Walsh table of real rational Chebyshev approximants on an interval. The same is true in complex CF (Carathéodory-Fejér) approximation on a circle. We show that these block structure results have a common origin in the existence of equioscillation-type characterization theorems for each of these three approximation problems. Consideration of position within a block is then shown to be a fruitful guide to various questions whose answers are affected by degeneracy.

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References

  1. [1]
    G. Baker and P. Graves-Morris, Padé Approximants (2 vols.), Encyc. of Math. v. 13 and 14, Addison-Wesley, 1981.Google Scholar
  2. [2]
    A. Bultheel, paper in this volume.Google Scholar
  3. [3]
    G. Claessens, On the structure of the Newton-Padé table, J. Approx. Theory 22 (1978), 304–319.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    M. A. Gallucci and W. B. Jones, Rational approximations corresponding to Newton series, J. Approx. Theory 17 (1976), 366–392.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    K. O. Geddes, Block structure in the Chebyshev-Padé table, SIAM J. Numer. Anal. 18 (1981), 844–861.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    W. Gragg, The Padé table and its relation to certain algorithms of numerical analysis, SIAM Review 14 (1972), 1–62.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    M. H. Gutknecht, On complex rational approximation II, in Computational Aspects of Complex Analysis, H. Werner et al. (eds.), D. Reidel, Dordrecht/Boston/Lancaster, 1983.Google Scholar
  8. [8]
    M. H. Gutknecht, E. Hayashi, and L. N. Trefethen, The CF table, in preparation.Google Scholar
  9. [9]
    M. H. Gutknecht and L. N. Trefethen, Nonuniqueness of best rational Chebyshev approximations on the unit disk, J. Approx. Theory 39 (1983), 275–288.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    A. Magnus, The connection between P-fractions and associated fractions, Proc. Amer. Math. Soc. 25 (1970), 676–679.MathSciNetMATHGoogle Scholar
  11. [11]
    A. Magnus, private communication, 1983.Google Scholar
  12. [12]
    G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer, 1967.Google Scholar
  13. [13]
    A. Ruttan, The length of the alternation set as a factor in determining when a best real rational approximation is also a best complex rational approximation, J. Approx. Theory 31 (1981), 230–243.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    L. N. Trefethen, Rational Chebyshev approximation on the unit disk, Numer. Math. 37 (1981), 297–320.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    L. N. Trefethen, Chebyshev approximation on the unit disk, in Computational Aspects of Complex Analysis, H. Werner et al. (eds.), D. Reidel, Dordrecht/Boston/Lancaster, 1983.Google Scholar
  16. [16]
    L. N. Trefethen and M. H. Gutknecht, On convergence and degeneracy in rational Padé and Chebyshev approximation, SIAM J. Math. Anal., to appear.Google Scholar
  17. [17]
    H. Werner, On the rational Tschebyscheff operator, Math. Zeit. 86 (1964), 317–326.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    H. Werner and L. Wuytack, On the continuity of the Padé operator, SIAM J. Numer. Anal. 20 (1983), 1273–1280.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Lloyd N. Trefethen
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew York

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