Weak copositive and interwining approximation
type: Item
instance of: scholarly article
title: Weak copositive and interwining approximation
zbMATH Open document ID: 0920.41006
published in: Journal of Approximation Theory
publication date: 1999-05-31T00:00:00Z
review text: The authors continue their research into the question of the degree of approximation of functions in \(L_p[0,1]\), by polynomials satisfying various constraints. If a function changes its sign finitely many times in \([0,1]\), the approximation by polynomials which keep the sign of the function throughout the interval is called copositive approximation. For a general function \(f\), the concept of one-sided approximation is that of approximating \(f\) by pairs of polynomials \(p_n\) and \(q_n\) so that \(p_n\leq f\leq q_n\) throughout the interval. Recently [Can. J. Math. 49, No. 1, 74-99 (1997; Zbl 0879.41006)], the authors have introduced a concept which is strongly related to the two notions described above. Namely, the concept of intertwining approximation which for a prescribed finite collection of points in \([0,1]\), considers the approximation of \(f\) by pairs of polynomials \(p_n\) and \(q_n\) so that both interpolate \(f\) at the prescribed points and both alternate above and below \(f\) as they pass through these points and keep \(f\) in between them at all time. That is, the polynomials \(p_n\) and \(q_n\) are such that, say, \[ \text{sgn}(-1)^j(p_n(x)-f(x))= \text{sgn}(-1)^j(f(x)-q_n(x)),\tag{*} \] in the \(j\)th subinterval formed by the prescribed collection of points. Here, following recent work of \textit{I. A. Shevchuk} and the reviewer [J. Approximation Theory 95, 53-81 (1998; Zbl 0911.41003)], they investigate the improvement that can be achieved in the degree of copositive and of intertwining approximation if we relax a bit the constraints by requiring in the case of copositive approximation, that the polynomials keep the sign of \(f\), except in small neighborhoods of the points where \(f\) changes its sign; and in the case of intertwining approximation, that the pairs of polynomials satisfy (*) except in small neighborhoods of the prescribed points. Yet another possibility which is a bit more restrictive is that the pairs will be intertwining…
reviewed by: D. Leviatan
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A29
zbMATH DE Number: 1271290
zbMATH Keywords: copositive approximation
zbMATH Keywords: intertwining approximation
zbMATH Keywords: degree of approximation
author: Yingkang Hu
author: Kirill A. Kopotun
author: Xiang Ming Yu
MaRDI profile type: Publication
cites work: Restricted Range Approximation by Splines and Variational Inequalities
cites work: Convex polynomial and spline approximation in \(L_ p, 0<p<\infty\)
cites work: Q4273944
cites work: Interpolation of Besov Spaces
cites work: Q4396599
cites work: Positive and copositive spline approximation in \(L_ p[0,1]\)
cites work: On positive and copositive polynomial and spline approximation in \(L_ p[-1,1],\quad 0<p<\infty\)
cites work: Constrained Approximation in Sobolev Spaces
cites work: Copositive polynomial approximation in C [0,1]
cites work: Copositive polynomial and spline approximation
cites work: The Degree of Copositive Approximation and a Computer Algorithm
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: A NOTE ON SIMULTANEOUS APPROXIMATION IN Lp[—1,1] (1≤<ρ<∞)
cites work: On copositive approximation by algebraic polynomials
cites work: The Degree of Copositive Approximation by Polynomials
cites work: The Rate of Monotone Spline Approximation in the $L_p $-Norm
cites work: Nearly comonotone approximation
cites work: Q3895980
cites work: Q4730231
cites work: Q4202368
description: scientific article; zbMATH DE number 1271290