Convex Polynomial Approximation in Lp (0 < p < 1)
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We prove that for each convex function ƒ ϵ Lp, 0 < p ≤ 1, there exists a convex algebraic polynomial Pn of degree ≤n such that [Formula presented] where ωΨ2(ƒ, t)p is the Ditzian-Totik modulus of smoothness of f(hook) in Lp, and C depends only on p. Moreover, if ƒ is also nondecreasing, then the polynomial Pn can also be taken to be nondecreasing, thus we have simultaneous monotone and convex approximation in this case. © 1993 Academic Press, Inc.
author list (cited authors)
Devore, R. A., & Leviatan, D.