On a global property of approximately differentiable functions
type: Item
instance of: scholarly article
title: On a global property of approximately differentiable functions
zbMATH Open document ID: 0629.26007
DOI: 10.1007/BF01137673
author: N. M. Isakov
published in: Mathematical Notes
publication date: 1987-01-01T00:00:00Z
review text: Let a jet of order \(r\) from a Borel set \(B\) into \(R^m\) denote the set of measurable functions \(\{f^a\}_{| a| \leq r}: B\to R^m,\) where \(a=(a_1,\ldots,a_n)\) is a multi-index. The main result of the paper says that if \(m(B)<\infty\), then the jet \(\{f^a\} \in \operatorname{ap} T^r(B,R^m)\) if and only if for each \(\varepsilon >0\) there exist a compact set \(K\subset B\) and a function \(F\in C^r(R^n,R^m)\) such that \(m(K)>m(B)-\varepsilon\) and \(f^a| K=D^aF| K\). This is an answer to the question formulated by \textit{H. Federer} in ``Geometric measure theory'' [Berlin: Springer Verlag (1969; Zbl 0176.00801), p. 229], and a generalization (for higher orders) of Th. 3.1.16 in Federer's book.
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 58A20
zbMATH DE Number: 4023603
zbMATH Keywords: approximately differentiable functions
zbMATH Keywords: approximately differentiable jets
zbMATH Keywords: measurable functions
reviewed by: Władysław Wilczyński
MaRDI profile type: Publication
cites work: Q5565773
cites work: Q3280291
cites work: Q3027681
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description: scientific article; zbMATH DE number 4023603