Advertisement

The strong uniqueness constant in complex approximation

  • T. J. Rivlin
Approximation And Interpolation Theory
  • 481 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 1105)

Abstract

Let f(x) belong to the set C(B) of continuous functions, possibly complex valued, on a compact subset, B, of a finite dimensional euclidean space. Let V be a finite dimensional subspace of C(B). v0 ε V is a strongly unique best approximation to f if (1) ‖f − v‖ ≥ ‖f − v0‖+γ ‖v − v0∥ holds for some γ(f, B, V) > 0 and all v ε V. The largest γ (= γ*) such that (1) holds is the strong uniqueness constant for f. The strong uniqueness constant has previously been determined in essentially one case, namely, the approximation of a monomial by lower degree polynomials in the real case. When f(z)=zn, B is the closed unit disc in the complex plane and V is the set of polynomials of degree at most n−1, γ*=1/n. We show that this fact is a trivial consequence of a simple result of Szász (1917). Some related results and problems are also discussed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bartelt, M. W., and H. W. McLaughlin, Characterizations of strong unicity in approximation theory, J. Approx. Theory 9 (1973), 255–266.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cline, A. K., Lipschitz conditions on uniform approximation operators, J. Approx. Theory 8 (1973), 160–172.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gutknecht, M. H., On complex rational approximation; Part I: The characterization problem, Computational Aspects of Complex Analysis (Eds. H. Werner, L. Wuytack, E. Ng, H. J. Bünger), D. Reidel, Dordrecht, Holland (1973), 79–101.Google Scholar
  4. 4.
    Newman, D. J., Polynomials and rational functions, Approximation Theory and Applications (Ed. Z. Ziegler), Academic Press, N. Y. (1981), 265–282.Google Scholar
  5. 5.
    Pólya, G., and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vol. II, Dover Publications, N. Y., 1945.zbMATHGoogle Scholar
  6. 6.
    Rivlin, T., The best strong uniqueness constant for a multivariate Chebyshev polynomial, J. Approx. Theory, to appear.Google Scholar
  7. 7.
    Szász, O., Über nichtnegative trigonometrische Polynome, Sitzungsb. d. Bayer. Akad. der Wiss. Math. — Phys. Kl., 1917, 307–320.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • T. J. Rivlin
    • 1
  1. 1.Mathematical Sciences DepartmentIBM Thomas J. Watson Research CenterYorktown HeightsUSA

Personalised recommendations