Bernstein type theorems for compact sets in \(\mathbb{R}^ n\) revisited
The classical Bernstein inequality asserts that for complex polynomials \(P\) of degree at most \(k\), we have \[ | P'(x)|\leq {k\over \sqrt{1- x^ 2}} \| P\|_{L_ \infty[-1, 1]},\quad x\in (-1, 1). \] In 1937 Bernstein established an improvement of this for real polynomials: \[ | P'(x)|\leq {k\over \sqrt{1- x^ 2}} (\| P\|^ 2_{L_ \infty[-1, 1]}- P^ 2(x))^{1/2},\quad x\in (-1, 1). \] In this interesting paper, the author continues his program to derive analogues of these for polynomials in several variables, over real or complex domains in \(\mathbb{R}^ n\) or \(\mathbb{C}^ n\). For example, if \(\|\cdot\|\) is a norm on \(\mathbb{R}^ n\), and if \(E:= \{\underline x: \|\underline x\|\leq 1\}\), then for real polynomials \(P\) of \(n\) variables that are of degree at most \(k\) in each variable, \[ | D_{\underline v} P(\underline x)|\leq k\| \underline v\| (1- \|\underline x\|^ 2)^{- 1/2} (\| P\|^ 2_{L_ \infty(E)}- P^ 2(x))^{1/2} \] provided \(\| \underline x\|< 1\). Here \(D_{\underline v}\) denotes the directional derivative in the direction \(\underline v\). Moreover, there is the Markov inequality \[ \| D_{\underline v} P\|_{L_ \infty(E)}\leq k^ 2 \| \underline v\|\;\| P\|_{L_ \infty(E)}. \] The author also provides estimates for more general open sets that involve the generalized Green's function of its complement, as well as estimates for grad P. This paper is essential reading for those interested in converse theorems of multivariate polynomial approximation.
scientific article; zbMATH DE number 701329