A note on convex approximation in \(L_ p\)
type: Item
instance of: scholarly article
title: A note on convex approximation in \(L_ p\)
zbMATH Open document ID: 0823.41015
author: Margarita Nikotjeva-Hedberg
author: Vladimir A. Operstein
published in: Journal of Approximation Theory
publication date: 1995-05-18T00:00:00Z
review text: This note provides estimates for the approximation of a bounded convex function \(f\in L_ p[- 1, 1]\), \(1\leq p\leq \infty\), by convex polynomials. The estimates involve the second Ditzian-Totik modulus of smoothness of \(f\) in the sup-norm, hence one reason for the requirement of boundedness of \(f\). With \(\varphi:= \sqrt{1- x^ 2}\) and \(q\) satisfying \({1\over q}+ {1\over p}= 1\), it shows in a constructive way, that for each \(n\geq 1\) a convex polynomial \(P_ n\) exists, such that \[ \| f- P_ n\|_ p\leq Cn^{- 2/p} \omega^ \varphi_ 2(f, 1/n)^{1/q}_ \infty, \] where \(C= C_ 0 \delta(f)^{1/p}\), \(C_ 0\) an absolute constant and \[ \delta(f):= \sup_{x\in [- 1,1]} f(x)- \inf_{x\in [- 1,1]} f(x). \] Note that if \(f\in C[- 1,1]\) (that is if \(f\) is also continuous at the end points \(\pm 1\)), then for \(1< p< \infty\) we get convex approximation with accuracy of \(o(n^{- 2/p})\). For \(p= 1\) we have \(O(n^{- 2})\) which is best possible. Better estimates are valid for \(p= \infty\).
Mathematics Subject Classification ID: 41A29
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 756648
zbMATH Keywords: degree of approximation
zbMATH Keywords: convex approximation
reviewed by: D. Leviatan
MaRDI profile type: Publication
full work available at URL: jath.1995.1039
OpenAlex ID: W2064492618
DOI: 10.1006/JATH.1995.1039
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description: scientific article; zbMATH DE number 756648