Inequalities of the Kolmogorov type for norms of Riesz derivatives of multivariate functions and some of their applications
type: Item
instance of: scholarly article
title: Inequalities of the Kolmogorov type for norms of Riesz derivatives of multivariate functions and some of their applications
zbMATH Open document ID: 1271.41004
published in: Journal of Mathematical Sciences (New York)
publication date: 2013-04-24T00:00:00Z
review text: New exact Kolmogorov-type inequalities are derived which estimate the \(L_\infty \)-norm of the Riesz derivative \(D^\alpha f\) of a function \(f\) defined on the \(m\)-dimensional Euclidean space \(\mathbb{R} ^m\;(0<\alpha <2-m/s,\;s>m/2)\) by means of \(\| \Delta f\|_{L_s}\) and \(\| f\|_{L_\infty }\), where \(\Delta \) is the Laplace operator. Moreover, the following problems are studied. (1) The approximation of an unbounded operator \(D^\alpha \) by bounded ones. (2) The optimal recovery of the operator \(D^\alpha \) on elements of the class \(\{ f: \| \Delta f\|_s\leq 1\}\) given with known error. (3) Necessary and sufficient conditions for the existence of a function\(f\) for which the given numbers are upper bounds of the moduli of its derivatives of corresponding orders and \(\Delta f\in L_s(\mathbb{R}^m)\).
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 26E99
zbMATH DE Number: 6156958
zbMATH Keywords: Landau-Kolmogorov-type inequalities
zbMATH Keywords: fractional derivative
zbMATH Keywords: Laplace operator
zbMATH Keywords: approximation of unbounded operators
zbMATH Keywords: recovery of operators by information with error
author: V. F. Babenko
author: Nataliia Parfinovych
reviewed by: D. K. Ugulava
MaRDI profile type: Publication
full work available at URL: s10958-012-1045-3
OpenAlex ID: W2004864778
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DOI: 10.1007/S10958-012-1045-3
description: scientific article; zbMATH DE number 6156958