Are the degrees of best (co)convex and unconstrained polynomial approximation the same?

Abstract

Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ² the set of convex functions f ∈ ℂ[−,1]. Also, let E _n (f) and E ⁽²⁾ _n (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ² by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E ⁽²⁾ _n (f), and Lorentz and Zeller proved that the inverse inequality E ⁽²⁾ _n (f) ≦ cE _n (f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ².

In this paper we prove, for every α > 0 and function f ∈ Δ², that

$$ \sup \{ n^\alpha E_n^{(2)} (f):n \in \mathbb{N}\} \leqq c(\alpha )\sup \{ n^\alpha E_n (f):n \in \mathbb{N}\} , $$

where c(α) is a constant depending only on α. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (−1,1) is also investigated. It turns out that there are substantial differences between the cases s≦ 1 and s ≧ 2.