Acta Mathematica Hungarica
Volume/Issue:
Volume 123: Issue 3
Are the degrees of best (co)convex and unconstrained polynomial approximation the same?
Authors:
K. Kopotun University of Manitoba Department of Mathematics Winnipeg Manitoba R3T 2N2 Canada

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D. Leviatan Tel Aviv University Raymond and Beverly Sackler School of Mathematical Sciences Tel Aviv 69978 Israel

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I. Shevchuk National Taras Shevchenko University of Kyiv Faculty of Mechanics and Mathematics 01033 Kyiv Ukraine

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Pages:
273–290
Online Publication Date:
27 Jan 2009
Publication Date:
01 May 2009
Article Category:
Research Article
DOI:
https://doi.org/10.1007/s10474-009-8111-4
Keywords:
(co)convex approximation by polynomials; degree of approximation; 41A10; 41A25; 41A29
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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 En(f) and En(2) (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ2 by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ En(2) (f), and Lorentz and Zeller proved that the inverse inequality En(2) (f) ≦ cEn(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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Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation		 1950
Volumes
per Year		 3
Issues
per Year		 6
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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