Summary
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 iff(ɛz) 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(ɛm + 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 ofe z on the unit disk is taken as a central computational example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
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)
Akhieser, N.I.: Theory of approximation. (Appendix D) New York: Ungar, 1956
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)
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)
Clark, D.: Hankel forms, Toeplitz forms and meromorphic functions. Trans. Amer. Math. Soc.134, 109–116 (1968)
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)
Gragg, W.B.: The Padé table and its relation to certain algorithms of numerical analysis. SIAM Rev.14, 1–62 (1972)
Gutknecht, M.: Carathéodory-Fejér approximation. (in press, 1981)
Gutknecht, M. H., Trefethen, L.N.: Recursive digital filter design by the Carathéodory-Fejér method. IEEE Trans. Acoust. Speech Signal Proc. (in press, 1981)
Gutknecht, M.H., Trefethen, L.N.: Real polynomial Chebyshev approximation by the Carathéodory-Fejér method. SIAM J. Numer. Anal. (in press, 1981)
Henrici, P.: Applied and computational complex analysis, Vol. 1, New York: Wiley, 1974
Henrici, P.: Fast Fourier methods in computational complex analysis. SIAM Rev.21, 481–527 (1979)
Hoffman, K.: Banach spaces of analytic functions. Englewood Cliffs, N.J.: Prentice-Hall, 1962
Klotz, V.: Gewisse rationale Tschebyscheff-Approximationen in der komplexen Ebene. J. Approximation Theory19, 51–60 (1977)
Kung, S.Y.: New fast algorithms for optimal model reduction. Proceedings, 1980 Joint Automatic Control Conference, San Francisco. New York: IEEE, 1980
Meinardus, G.: Approximation of functions: theory and numerical methods. Berlin-Heidelberg-New York: Springer, 1967
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)
Schur, I.: deÜber Potenzreihen, die im Innern des Einheitskreises beschränkt sind. J. Reine Angew. Math.148, 122–145 (1918)
Silverman, L.M., Bettayeb, M.: Optimal approximation of linear systems. IEEE Trans. Automatic Control (in press, 1981)
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, 1976
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)
Trefethen, L.: Near-circularity of the error curve in complex Chebyshev approximation. J. Approximation Theory (in press, 1981)
Walsh, J.L.: Padé approximants as limits of rational functions of best approximation. J. Math. Mech.12, 305–312 (1964)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Trefethen, L.N. Rational Chebyshev approximation on the unit disk. Numer. Math. 37, 297–320 (1981). https://doi.org/10.1007/BF01398258
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01398258