On the Bernstein-Nikolsky inequality
type: Item
instance of: scholarly article
title: On the Bernstein-Nikolsky inequality
zbMATH Open document ID: 0803.41017
author: Ha Huy Bang
author: Mitsuo Morimoto
published in: Tokyo Journal of Mathematics
publication date: 1992-06-25T00:00:00Z
review text: Let \(1 \leq p < \infty\) and \(\sigma = (\sigma_ 1, \sigma_ 2, \dots, \sigma_ n)\), \(\sigma_ i > 0\), \(i=1,2, \dots,n\). Write \(M_{\sigma,p}\) for the space of entire functions of exponential type \(\sigma\) that belong to \(L^ p (\mathbb{R}^ n)\). The authors extend the Bernstein-Nikol'skij inequality by showing that \(\lim_{| \alpha | \to \infty} \sigma^{ - \alpha} \| D^ \alpha f \|_ p = 0\) for \(f \in M_{\sigma,p}\), where \(\alpha = (\alpha_ 1, \alpha_ 2, \dots, \alpha_ n)\) with \(\alpha_ i \geq 0\) for \(i = 1,2, \dots, n\). They also obtain Bernstein-Nikol'skij-type results for the directional derivatives of entire functions of exponential type.
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 26E05
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 13499
zbMATH Keywords: Bernstein-Nikol'skij inequality
zbMATH Keywords: directional derivatives
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270130503
OpenAlex ID: W2021754977
DOI: 10.3836/TJM/1270130503
description: scientific article; zbMATH DE number 13499