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  • Author or Editor: K. Kopotun x
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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 n (2) (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ2 by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E n (2) (f), and Lorentz and Zeller proved that the inverse inequality E n (2) (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

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sup \{ n^\alpha E_n^{(2)} (f):n \in \mathbb{N}\} \leqq c(\alpha )\sup \{ n^\alpha E_n (f):n \in \mathbb{N}\} ,$$ \end{document}
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.

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The paper deals with approximation of a continuous function, on a finite interval, which changes convexity finitely many times, by algebraic polynomials which are coconvex with it. We give final answers to open questions concerning the validity of Jackson type estimates involving the weighted Ditzian-Totik moduli of smoothness.

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