Bernstein type theorems for compact sets in \(\mathbb{R}^ n\) revisited
type: Item
instance of: scholarly article
title: Bernstein type theorems for compact sets in \(\mathbb{R}^ n\) revisited
zbMATH Open document ID: 0819.41013
author: Mirosław Baran
published in: Journal of Approximation Theory
publication date: 1994-12-11T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 701329
zbMATH Keywords: multivariate polynomial approximation
zbMATH Keywords: Bernstein inequality
zbMATH Keywords: Markov inequality
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
full work available at URL: jath.1994.1124
OpenAlex ID: W2017028065
DOI: 10.1006/JATH.1994.1124
description: scientific article; zbMATH DE number 701329