Markov-Bernstein type inequalities for multivariate polynomials on sets with cusps
type: Item
instance of: scholarly article
title: Markov-Bernstein type inequalities for multivariate polynomials on sets with cusps
zbMATH Open document ID: 0953.41008
author: András Kroó
author: Jószef Szabados
published in: Journal of Approximation Theory
publication date: 2001-01-22T00:00:00Z
review text: In the paper, the authors study the \(n\)-th order \textit{local Markov-Bernstein factor} \(M_n(K,x)\) = \(:=\sup\{\|\text{grad} p(x)\|:\;p\in\mathcal P^m_n,\;\|p\|_{C(K)}\leq 1\}\) at a \(w\)-cuspidal point \(x\) of the boundary of a compact subset \(K\) of \(\mathbb{R}^m\) such that \(K=\)cl(int \(K\)), where \(w\) is a \(C^1\)-function defined on \([0,1]\) such that \(0=w(0)<w(t),\;t\in(0,1]\) and \(w(t)/t\) is monotone increasing on (0,1], \(\mathcal P_n^m\) being the space of all polynomials of degree at most \(n\) in \(m\) variables. In particular, they provide asymptotically sharp bounds for polynomial cusps. They also introduce the notion of \textit{inner cusps} and show that for them the rates of the local Markov-Bernstein factor improve substantially. \textit{Reviewer's remark}. The authors are motivated by the notion of (globally) \textit{uniformly polynomially cuspidal} sets introduced by \textit{W. Pawłucki} and \textit{W. Pleśniak} [Math. Ann. 275, 467-480 (1986; Zbl 0591.32016)] in connection with the construction of a continuous linear operator extending \(C^\infty\)-Whitney jets on \(K\) to \(C^\infty\)-functions on \(\mathbb{R}^m\) (cf \textit{W. Pawłucki}, \textit{W. Plesniak} [Studia Math. 88, No. 3, 279-287 (1988; Zbl 0778.26010)]). They remark (on page 73) that ``estimates for \(M_n(K,x)\) can be easily extended to uniform MB-factors defined as \(M_n(K):=\sup_{x\in K}M_n(K,x)\) by imposing the needed geometric conditions uniformly for every \(x\in K\)''. However, such a claim may provide ambiguity. Consider e.g. the set \(K=E\cup\{(0,0)\}\subset\mathbb{R}^2\), where \(E \{(x,y)\in\mathbb{R}:\;0<x\leq 1,\;0<y\leq \exp(-1/x)\}\). Then for each \((x,y)\in E\), \(M_n(K,(x,y))\leq \text{const}(x,y) n^2\), while by \textit{M. Zerner} [C. R. Acad. Sci., Paris, Ser. A 268, 218-220 (1969; Zbl 0189.14601)], for any \(r>0\), \(\lim_{n\to\infty}n^{-r}M_n(K,(0,0))=\infty\). Actually, for application, the knowledge of the local MB factor is less useful than …
reviewed by: Wiesław Pleśniak
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 1417552
zbMATH Keywords: multivariate polynomials
zbMATH Keywords: derivatives
zbMATH Keywords: domains with cusps
zbMATH Keywords: polynomial curves
zbMATH Keywords: Markov inequality
zbMATH Keywords: Bernstein inequality
MaRDI profile type: Publication
full work available at URL: jath.1999.3378
OpenAlex ID: W2066902620
cites work: Markov inequality on sets with polynomial parametrization
cites work: Q4858543
cites work: Remez-Type Inequalities on the Size of Generalized Polynomials
cites work: Inégalité de Markov dans les ensembles effiles
cites work: Markov's inequality and \(C^{\infty}\) functions on sets with polynomial cusps
cites work: Q3718964
cites work: Q4852310
DOI: 10.1006/JATH.1999.3378
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description: scientific article; zbMATH DE number 1417552