One example in monotone approximation
type: Item
instance of: scholarly article
title: One example in monotone approximation
zbMATH Open document ID: 0868.41015
author: I. O. Shevchuk
published in: Journal of Approximation Theory
publication date: 1996-10-27T00:00:00Z
review text: It is readily seen that a monotone function \(f\in C^1[-1,1]\) can be approximated by monotone polynomials of degree not exceeding \(n\) at the rate \(E_{n-1}(f')\), where \(E_n(g)\) is the degree of unconstrained approximation of \(g\in C[-1,1]\) by polynomials of degree \(\leq n\). It is also quite easy and well-known that for \(f\in C^1[-1,1]\), we have the estimate \(E_n(f)\leq{C\over n}E_{n-1}(f')\). It has been known for many years that the degree of monotone approximation of a monotone function is not as small as the degree of unconstrained approximation. But it is quite natural to ask if we can still improve the estimate for monotone approximation, for instance by having an estimate of the type \(n^{-\alpha}E_{n-1}(f')\), for some \(\alpha>0\). The author shows that this is impossible by proving that for every \(n>0\) there is a nondecreasing \(f_n\in C^1[-1,1]\) such that \[ {1\over 200}< {E^{(1)}_n(f)\over E_{n-1}(f')}< 2. \] Here \(E^{(1)}_n(f)\) denote the degree of monotone approximation by polynomials of degree \(\leq n\).
Mathematics Subject Classification ID: 41A29
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 938741
zbMATH Keywords: monotone approximation
reviewed by: D. Leviatan
MaRDI profile type: Publication
full work available at URL: jath.1996.0068
OpenAlex ID: W2075258791
DOI: 10.1006/JATH.1996.0068
description: scientific article; zbMATH DE number 938741