Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T13:30:32.281Z Has data issue: false hasContentIssue false
  • English
  • Français

Shape-preserving polynomial approximation in C[– 1, 1]

Published online by Cambridge University Press:  24 October 2008

Z. Ditzian ,
D. Jiang  and
D. Leviatan
Z. Ditzian
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, CanadaT6G 2G1
D. Jiang
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, CanadaT6G 2G1
D. Leviatan
Affiliation:
Department of Mathematics, Raymond and Beverly Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Get access Rights & Permissions [Opens in a new window]

Abstract

A recent result of the first two authors has bridged the gap between the Timantype pointwise estimate and the norm estimate for polynomial approximation in C[– 1, 1]. Here we show that, when considering the second order modulus of smoothness, the aforementioned result holds with polynomials that preserve monotonicity, convexity and the values of the function at the end points of the interval [–1, 1]. This yields a generalization of results by Telyakovskii, Gopengauz, DeVore, Yu and the third author. Moreover, the construction here does not depend on the construction for the non-constrained approximation by polynomials.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 2 , September 1992 , pp. 309 - 316
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]DeVore, R. A., Leviatan, D. and Yu, X. M.. Polynomial approximation in L p(O < p < 1). Constr. Approx., to appear.Google Scholar
[2]DeVore, R. A. and Yu, X. M.. Pointwise estimates for monotone polynomial approximation. Constr. Approx. 1 (1985), 323–331.CrossRefGoogle Scholar
[3]Ditzian, Z. and Jiang, D.. Approximation of functions by polynomials in C[–1, 1]. Canad. J. Math., to appear.Google Scholar
[4]Ditzian, Z. and Totik, V.. Moduli of Smoothness (Springer-Verlag, 1987).CrossRefGoogle Scholar
[5]Leviatan, D.. Pointwise estimates for convex polynomial approximation. Proc. Amer. Math. Soc. 98 (1986), 471–474.CrossRefGoogle Scholar
[6]Leviatan, D.. Monotone and comonotone polynomial approximation revisited. J. Approx. Theory 53 (1988), 1–16.CrossRefGoogle Scholar
[7]Lorentz, C. O.. Approximation of Functions (Holt, Rinehart and Winston, 1966; Chelsea, 1985).Google Scholar
[8]Timan, A. F.. Theory of Approximation of Functions of a Real Variable (Pergamon, 1963).Google Scholar