Pointwise estimates for higher order convexity preserving polynomial approximation

Abstract

Abstract 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 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 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.