On monotone and convex approximation by algebraic polynomials

Abstract

The following results are obtained: If α>0, α≠2, α$\bar \in$ [3, 4], andf is a nondecreasing (convex) function on [−1, 1] such thatE _n (f) ≤n ^−α for any n>α, then E ⁽¹⁾_n (f)≤Cn ^−α (E ⁽²⁾_n (f)≤Cn ^−α) for n>α, where C=C(α), E_n(f) is the best uniform approximation of a continuous function by polynomials of degree (n−1), and E ⁽¹⁾_n (f) (E ⁽²⁾_n (f)) are the best monotone and convex approximations, respectively. For α=2 (α ∈ [3, 4]), this result is not true.