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Pointwise estimates for higher order convexity preserving polynomial approximation

Published online by Cambridge University Press:  17 February 2009

Jia-Ding Cao  and
Heinz H. Gonska
Jia-Ding Cao
Affiliation:
Dept of Mathematics, Fudan University, Shanghai, PRC.
Heinz H. Gonska
Affiliation:
Dept of Mathematics, University of Duisburg, D-47048 Duisburg, Germany.
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Abstract

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DeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 2 , October 1994 , pp. 213 - 233
Copyright
Copyright © Australian Mathematical Society 1994

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