Convex Polynomial Approximation in Lp (0 < p < 1) Original Research Article

We prove that for each convex function ƒ ∈ Lp, 0 < p < 1, there exists a convex algebraic polynomial Pn of degree ≤n such that [formula] where ωφ2(ƒ, t)p is the Ditzian-Totik modulus of smoothness of ƒ 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.