Numerische Mathematik

, Volume 37, Issue 2, pp 297–320 | Cite as

Rational Chebyshev approximation on the unit disk

  • Lloyd N. Trefethen
A numerical Method for Computing Turning Points. II. Numerical Analysis of a Nonlinear Boundary Value Problem


In a recent paper we showed that error curves in polynomial Chebyshev approximation of analytic functions on the unit disk tend to approximate perfect circles about the origin [23]. Making use of a theorem of Carathéodory and Fejér, we derived in the process a method for calculating near-best approximations rapidly by finding the principal singular value and corresponding singular vector of a complex Hankel matrix. This paper extends these developments to the problem of Chebyshev approximation by rational functions, where non-principal singular values and vectors of the same matrix turn out to be required. The theory is based on certain extensions of the Carathéodory-Fejér result which are also currently finding application in the fields of digital signal processing and linear systems theory.

It is shown among other things that iffz) is approximated by a rational function of type (m, n) for ɛ>0, then under weak assumptions the corresponding error curves deviate from perfect circles of winding numberm+n+1 by a relative magnitudeO m + n + 2 as ɛ→0. The “CF approximation” that our method computes approximates the true best approximation to the same high relative order. A numerical procedure for computing such approximations is described and shown to give results that confirm the asymptotic theory. Approximation ofez on the unit disk is taken as a central computational example.

Subject Classifications

AMS(MOS) 30D50 30E10 41A50 


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  1. 1.
    Adamian, V.M., Arov, D.Z., Krein, M.G.: Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem. Math. USSR Sb.15, 31–73 (1971)Google Scholar
  2. 2.
    Akhieser, N.I.: On a minimum problem in the theory of functions and on the number of roots of an algebraic equation which lie inside the unit circle. Izv. Akad. Nauk. SSSR, Otd. Mat. Estestv. Nauk9, 1169–1189 (1931) (in Russian)Google Scholar
  3. 3.
    Akhieser, N.I.: Theory of approximation. (Appendix D) New York: Ungar, 1956Google Scholar
  4. 4.
    Bultheel, A., de Wilde, P.: On the Adamian-Arov-Krein approximation, identification and balanced realization of a system. IEEE Trans. Circuits and Systems (in press, 1981)Google Scholar
  5. 5.
    Carathéodory, C., Fejér, L.: Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz. Rend. Circ. Mat. Palermo32, 218–239 (1911)Google Scholar
  6. 6.
    Clark, D.: Hankel forms, Toeplitz forms and meromorphic functions. Trans. Amer. Math. Soc.134, 109–116 (1968)Google Scholar
  7. 7.
    Genin, Y.V., Kung, S.Y.: A two-variable approach to the model reduction problem with Hankel norm criterion. IEEE Trans. Circuits and Systems (in press, 1981)Google Scholar
  8. 8.
    Gragg, W.B.: The Padé table and its relation to certain algorithms of numerical analysis. SIAM Rev.14, 1–62 (1972)Google Scholar
  9. 9.
    Gutknecht, M.: Carathéodory-Fejér approximation. (in press, 1981)Google Scholar
  10. 10.
    Gutknecht, M. H., Trefethen, L.N.: Recursive digital filter design by the Carathéodory-Fejér method. IEEE Trans. Acoust. Speech Signal Proc. (in press, 1981)Google Scholar
  11. 11.
    Gutknecht, M.H., Trefethen, L.N.: Real polynomial Chebyshev approximation by the Carathéodory-Fejér method. SIAM J. Numer. Anal. (in press, 1981)Google Scholar
  12. 12.
    Henrici, P.: Applied and computational complex analysis, Vol. 1, New York: Wiley, 1974Google Scholar
  13. 13.
    Henrici, P.: Fast Fourier methods in computational complex analysis. SIAM Rev.21, 481–527 (1979)Google Scholar
  14. 14.
    Hoffman, K.: Banach spaces of analytic functions. Englewood Cliffs, N.J.: Prentice-Hall, 1962Google Scholar
  15. 15.
    Klotz, V.: Gewisse rationale Tschebyscheff-Approximationen in der komplexen Ebene. J. Approximation Theory19, 51–60 (1977)Google Scholar
  16. 16.
    Kung, S.Y.: New fast algorithms for optimal model reduction. Proceedings, 1980 Joint Automatic Control Conference, San Francisco. New York: IEEE, 1980Google Scholar
  17. 17.
    Meinardus, G.: Approximation of functions: theory and numerical methods. Berlin-Heidelberg-New York: Springer, 1967Google Scholar
  18. 18.
    Motzkin, T.S., Walsh, J.L.: Zeros of the error function for Tchebycheff approximation in a small region. Proc. London Math. Soc.3, 90–98 (1963)Google Scholar
  19. 19.
    Schur, I.: deÜber Potenzreihen, die im Innern des Einheitskreises beschränkt sind. J. Reine Angew. Math.148, 122–145 (1918)Google Scholar
  20. 20.
    Silverman, L.M., Bettayeb, M.: Optimal approximation of linear systems. IEEE Trans. Automatic Control (in press, 1981)Google Scholar
  21. 21.
    Smith, B.T., Boyle, J.M., Dongarra, J.J., Garbow, B.S., Ikebe, Y., Klema, V.C., Moler, C.B.: Matrix eigensystem routines — EISPACK guide. (Lecture Notes in Computer Sciences. Vol. 6, 2nd ed.) Berlin-Heidelberg-New York: Springer, 1976Google Scholar
  22. 22.
    Takagi, T.: On an algebraic problem related to an analytic theorem of Carathéodory and Fejér. Japan J. Math.1, 83–93 (1924) and2, 13–17 (1925)Google Scholar
  23. 23.
    Trefethen, L.: Near-circularity of the error curve in complex Chebyshev approximation. J. Approximation Theory (in press, 1981)Google Scholar
  24. 24.
    Walsh, J.L.: Padé approximants as limits of rational functions of best approximation. J. Math. Mech.12, 305–312 (1964)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Lloyd N. Trefethen
    • 1
  1. 1.Computer Science DepartmentStanford UniversityStanfordUSA

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