Markov-Bernstein type inequalities for multivariate polynomials on sets with cusps
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 that of the global one (\textit{see e.g.} the reviewer [J. Approximation Theory 61, No. 1, 106-117 (1990; Zbl 0702.41023)]).
scientific article; zbMATH DE number 1417552