The degree of coconvex polynomial approximation

Let f S ∈ C[-1,1] change its convexity finitely many times in the interval, say s times, at Y_s : -1 < y₁ < ⋯ < y_s < 1. We estimate the degree of approximation of f by polynomials of degree n, which change convexity exactly at the points Y_s. We show that provided n is sufficiently large, depending on the location of the points Y_s, the rate of approximation is estimated by the third Ditzian-Totik modulus of smoothness of f multiplied by a constant C(s), which depends only on s.