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Convex Polynomial Approximation in the Uniform Norm: Conclusion

Published online by Cambridge University Press:  20 November 2018

K. A. Kopotun ,
D. Leviatan  and
I. A. Shevchuk
K. A. Kopotun
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, e-mail: [email protected]
D. Leviatan
Affiliation:
School of Mathematical Sciences, Raymond and Beverley Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel, e-mail: [email protected]
I. A. Shevchuk
Affiliation:
Faculty of Mechanics and Mathematics, National Taras Shevchenko University of Kyiv, Kyiv 01033, Ukraine, e-mail: [email protected]
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Abstract

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Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each $r$-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian–Totik $k$th modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs $(k,\,r)$ we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

Keywords

41A1041A2541A29
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 6 , 01 December 2005 , pp. 1224 - 1248
Copyright
Copyright © Canadian Mathematical Society 2005

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