Bernstein type inequalities on star-like domains in $${\mathbb{R }}^d$$ with application to norming sets

Abstract

Let \(P^d_n\) be the space of real algebraic polynomials of \(d\) variables and degree at most \(n, \, K\subset \mathbb{R }^{d}\) a compact set, \(||p||_K:=\sup _{\mathbf{x} \in K}|p(\mathbf{x})|\) the usual supremum norm on \(K\). Let \(\varphi _K(\mathbf{x}):=\inf \{\alpha >0:\mathbf{x}/\alpha \in K\}\) denote the Minkowski functional of \(K\). In this note we shall prove that if \(K\) is a star-like domain with Lip\(\alpha \) boundary, that is \(\varphi _K(\mathbf{x})\) satisfies the Lip\(\alpha \) condition, \(0<\alpha \le 1\) then the following Bernstein type inequality holds: for any \(p\in P^d_n, \Vert p\Vert _K=1\) and \(\mathbf{x}\in \mathrm{Int}K\) $$\begin{aligned} |\nabla p|(\mathbf{x})\le \frac{cn}{(1-\varphi _K(\mathbf{x}))^{\frac{1}{\alpha }-\frac{1}{2}}}, \end{aligned}$$ where \(|\nabla p|\) stands for the Euclidean length of the gradient of \(p\). Furthermore, if \(1<\alpha \le 2\) and \(K\) is a \(C^\alpha \) star like-domain, that is \(\nabla \varphi _K(\mathbf{x})\) has the Lip\((\alpha -1)\) property, then the same inequality holds for the tangential derivatives of \(p\). These new Bernstein type inequalities are applied for the study of cardinality of norming sets, or admissible meshes. The sequence of discrete sets \(\mathbf{Y}=\{Y_n\subset K, n\in \mathbb{N }\}\) is called an optimal admissible mesh in \(K\) if there exist constants \(c_1, c_2\) depending only on \(K\) such that $$\begin{aligned} ||p||_K\le c_1||p||_{Y_n},\quad p\in P^d_n, n\in \mathbb{N }, \end{aligned}$$ and \(card(Y_n)\le c_2n^d, n\in \mathbb{N }\). It was proved earlier that optimal admissible meshes exist in \(C^{2}\) star-like domains. In this paper it will be shown that \(C^{2-\frac{2}{d}}\) smoothness also suffices for their existence.