J Austral Math Soc Ser B 36 pp213--233, 1994.

Pointwise estimates for higher order convexity preserving polynomial approximation

Jia-Ding Cao and Heinz H. Gonska

Abstract

De Vore-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 H_{n, 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 H_{n, s, j}, show that they basically have the same shape preservation behavior while interpolating at the endpoints of [-1, 1], and also satisfy Telyakovskii- and De Vore-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, De Vore, Yu and Leviatan.

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Authors

Jia-Ding Cao

Dept of Mathematics, Fudan University, Shanghai, PRC.

Heinz H. Gonska

Dept of Mathematics, University of Duisburg, D-47048 Duisburg, Germany.