A Note on Convex Approximation in Lp

A convex function f given on [−1, 1] can be approximated in Lp, 1 < p < ∞, by convex polynomials Pn of degree at most n with the accuracy o(n−2/p). This follows from the estimate ∥f−Pn∥p ≤ c · n−2/p·ωφ2(f, n−1)1/q, where 1 ≤ p ≤ ∞, p−1 + q-−1 = 1, φ(x) = (1 − x2)1/2, and ωφ2(f, t) is the Ditzian-Totik modulus of smoothness in the uniform metric.