Some questions of the theory of approximatively differentiable functions
type: Item
instance of: scholarly article
title: Some questions of the theory of approximatively differentiable functions
zbMATH Open document ID: 0642.46040
DOI: 10.4171/ZAA/252
author: N. M. Isakov
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Omega\) be a Borel subset in \(R^ n\) and \(\mu\) is a regular Borel measure on \(\Omega\) such that \(\mu (\Omega)<\infty\). Let f be a mapping from \(\Omega\) into \(R^ m\) such that \(| f| \in L_{\infty}(\Omega,\mu)\). Then for any positive real number \(\epsilon\) there exist \(g\in C(R^ n,R^ m)\) and a compact subset K of \(\Omega\) such that \(\mu(K)>\mu (\Omega)-\epsilon\) and \(f|_ K=g|_ K\). The author considered the case in which we can choose \(g\in C^ r(R^ n,R^ m).\) Using the Whitney extension theorem he reduced the problem to consider some special classes of functions. The author also obtained a right inverse mapping theorem for these classes and extended the results to the case in which \(R^ m\) is replaced by a normed space.
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 58C20
Mathematics Subject Classification ID: 26B05
zbMATH DE Number: 4046409
zbMATH Keywords: approximation by smooth functions
zbMATH Keywords: Lusin measurable function
zbMATH Keywords: Borel measure
zbMATH Keywords: Whitney extension theorem
zbMATH Keywords: right inverse mapping theorem
reviewed by: Duong Minh Duc
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4046409