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• Author or Editor: K. Kopotun
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## Comonotone Polynomial Approximation in Lp[-1, 1], 0 < p ≦ ∞

Acta Mathematica Hungarica
Authors:
K. Kopotun
and
D. Leviatan
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## Are the degrees of best (co)convex and unconstrained polynomial approximation the same?

Acta Mathematica Hungarica
Authors:
K. Kopotun
,
D. Leviatan
, and
I. Shevchuk

## 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 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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## Coconvex approximation in the uniform norm: the final frontier

Acta Mathematica Hungarica
Authors:
K. Kopotun
,
D. Leviatan
, and
I. A. Shevchuk

## Summary

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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