Convex polynomial and spline approximation inLp, O
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We prove that a convex function f ∈ Lp[-1, 1], 0 < p < ∞, can be approximated by convex polynomials with an error not exceeding C ωφ3 (f, 1/n)p where ωφ3 (f, ·) is the Ditzian-Totik modulus of smoothness of order three of f. We are thus filling the gap between previously known estimates involving ωφ2 (f, 1/n)p, and the impossibility of having such estimates involving ω4. We also give similar estimates for the approximation of f by convex C0 and C1 piecewise quadratics as well as convex C2 piecewise cubic polynomials.
author list (cited authors)
DeVore, R. A., Hu, Y. K., & Leviatan, D.