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

Abstract

Let [-1,1] be the space of continuous functions on [-,1], and denote by Δ2 the set of convex functions f [-,1]. Also, let E n (f) and E n(2) (f) denote the degrees of best unconstrained and convex approximation of f Δ2 by algebraic polynomials of degree 0 and function f Δ2, that ⊃ α En(2) (f):n q c(α)⊃ α En (f):n 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 1 and s 2. © 2009 Springer Science+Business Media B.V.