Coconvex polynomial approximation
type: Item
instance of: scholarly article
title: Coconvex polynomial approximation
zbMATH Open document ID: 1021.41006
DOI: 10.1016/S0021-9045(02)00045-X
author: I. O. Shevchuk
author: D. Leviatan
published in: Journal of Approximation Theory
publication date: 2003-05-25T00:00:00Z
review text: Let \(f\in\mathbb{C}[-1,1]\) change its convexity finitely many times, in the interval. The authors are interested in estimating the degree of approximation of \(f\) by polynomials, and by piecewise polynomials, which are coconvex with it, namely, polynomials and piecewise polynomials that change their convexity exactly at the points where \(f\) does. The authors obtain Jackson-type estimates and summarize the positive and negative results in a truth-table. They prove two theorems. Theorem 4.1. If \(f\in\Delta^2(Y_s)\cap\mathbb{C}^2\), then for each \(k\leq 3\), \[ E^{(2)}_n(f,Y_s)\leq\frac c{n^2}\omega^\varphi_k(f'',1/n) \leq\frac c{n^2}\omega_k(f'',1/n),\quad n\geq N , \] where \(c=c(s)\) and \(N=N(Y_s)\). Furthermore, if \(s=1\) and \(k\leq 2\), then \(N=k+1\). Theorem 4.2. Let \(r\geq 3\) and assume that \(f\in\Delta^2(Y_s)\cap\mathbb{C}^r\). Then \[ E^{(2)}_n(f,Y_s)\leq\frac c{n^r}\omega^\varphi_k(f^{(r)},1/n),\quad n\geq N , \tag{1} \] holds for each \(k\geq 1\), with constants \(c=c(k,r,s)\) and \(N=N(k,r,Y_s)\). Furthermore, if \(s=1\), then \((1)\) holds with \(N=k+r-1\).
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 1915646
zbMATH Keywords: coconvex polynomial approximation
zbMATH Keywords: Jackson estimates
MaRDI profile type: Publication
cites work: Q3933398
cites work: Q3817079
cites work: Q4396599
cites work: Q5310459
cites work: Convex polynomial and spline approximation in \(C[-1,1]\)
cites work: Uniform estimates of the coconvex approximation of functions by polynomials
cites work: Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials
cites work: Uniform estimates of monotone and convex approximation of smooth functions
cites work: The degree of coconvex polynomial approximation
cites work: Pointwise Estimates for Convex Polynomial Approximation
cites work: Shape-preserving approximation by polynomials
cites work: Nearly comonotone approximation
cites work: Some positive results and counterexamples in comonotone approximation. II
cites work: Q4699399
cites work: Coconvex approximation.
cites work: Piecewise coapproximation and the Whitney inequality.
cites work: APPROXIMATION OF MONOTONE FUNCTIONS BY MONOTONE POLYNOMIALS
cites work: Q4369140
cites work: Q3913213
cites work: A counterexample in comonotone approximation in $L^p$ space
full work available at URL: s0021-9045(02)00045-x
OpenAlex ID: W2004233367
description: scientific article; zbMATH DE number 1915646