Bernstein inequality for differential operators
type: Item
instance of: scholarly article
title: Bernstein inequality for differential operators
zbMATH Open document ID: 0666.41011
DOI: 10.1007/BF01909364
published in: Analysis Mathematica
publication date: 1986-01-01T00:00:00Z
review text: Some analogues to Bernstein inequality for algebraic (with corresponding weights) and hyperspherical polynomials in \(L^ p\)-norms (1\(\leq p\leq \infty\), \(L_{\infty}=C)\) replacing the differential operator \(d^ 2/dx^ 2\) by corresponding differential operators D of second order are established. These inequalities have the form \((*)\quad \| DP_ n\|_{L_ p}\leq Const\cdot n^{1+\tau}\| P_ n\|_{L_ p},\) \(1\leq p\leq \infty\). In the case of algebraic polynomials D is the operator the eigenfunctions of which are the classical orthogonal polynomials of Jacobi (in this case \(\tau =1)\), Laguerre and Hermite (in these cases \(\tau =0)\). In case of m-dimensional (m\(\geq 2)\) spherical polynomials D is the Laplace-Beltrami operator and in this case \(\tau =1.If\) in the last case \(X=L_{\infty}\), the estimate (*) is made more precise. Namely, Const is replaced by (m-1) and the obtained inequality is exact.
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 4090093
zbMATH Keywords: Bernstein inequality
zbMATH Keywords: weights
zbMATH Keywords: hyperspherical polynomials
zbMATH Keywords: algebraic polynomials
author: Ar. S. Dzhafarov
reviewed by: D. K. Ugulava
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01909364
OpenAlex ID: W1758199069
description: scientific article; zbMATH DE number 4090093