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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bartelt, M. W., and H. W. McLaughlin, Characterizations of strong unicity in approximation theory, J. Approx. Theory 9 (1973), 255–266.
Cline, A. K., Lipschitz conditions on uniform approximation operators, J. Approx. Theory 8 (1973), 160–172.
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.
Newman, D. J., Polynomials and rational functions, Approximation Theory and Applications (Ed. Z. Ziegler), Academic Press, N. Y. (1981), 265–282.
Pólya, G., and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vol. II, Dover Publications, N. Y., 1945.
Rivlin, T., The best strong uniqueness constant for a multivariate Chebyshev polynomial, J. Approx. Theory, to appear.
Szász, O., Über nichtnegative trigonometrische Polynome, Sitzungsb. d. Bayer. Akad. der Wiss. Math. — Phys. Kl., 1917, 307–320.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 Springer-Verlag
About this paper
Cite this paper
Rivlin, T.J. (1984). The strong uniqueness constant in complex approximation. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072407
Download citation
DOI: https://doi.org/10.1007/BFb0072407
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13899-0
Online ISBN: 978-3-540-39113-5
eBook Packages: Springer Book Archive