On stability of classes of affine mappings
type: Item
instance of: scholarly article
title: On stability of classes of affine mappings
zbMATH Open document ID: 0865.30025
DOI: 10.1007/BF02112534
author: A. A. Egorov
published in: Siberian Mathematical Journal
publication date: 1997-06-30T00:00:00Z
review text: A class \(\mathcal A\) consists of all affine mappings \(g:\mathbb{R}^n\to\mathbb{R}^m\) of the form \(g(x)=\alpha(x)+b\), \(x\in\mathbb{R}^n\), restricted to a domain \(\Delta\subset\mathbb{R}^n\), where \(b\in\mathbb{R}^m\), \(\alpha\) belongs to a given set \(A\) of the space \(L(\mathbb{R}^n,\mathbb{R}^m)\) of linear mappings of \(\mathbb{R}^n\) into \(\mathbb{R}^m\), with the operator norm \(|\lambda|_{n,m}=\sup_{|x|\leq 1}|\lambda(x)|\), \(\lambda\in L(\mathbb{R}^n,\mathbb{R}^m)\) and represents the differential \(Dg\) of the affine mapping \(g\). Let be \(f:\Delta\to\mathbb{R}^m\), the ball \(B(x,r)\subset\Delta\) and let us denote \[ \omega_B(f,{\mathcal G}):=\inf_{g:B\to\mathbb{R}^n,g\in{\mathcal G}}\Biggl\{r^{-1}\sup_{y\in B}|f(y)-g(y)|\Biggr\}, \] \[ \Omega(f,{\mathcal G}):=\sup_{x\in\Delta} \Omega(x,f,{\mathcal G})=\sup_{x\in\Delta} \Biggl\{\varlimsup_{r\to 0} \omega_{B(x,r)}(f,{\mathcal G})\Biggr\}. \] Starting from \textit{F. Jones's} [Commun. Pure Appl. Math. 14, 391-413 (1961; Zbl 0102.17404); ibid. 21, 77-110 (1968; Zbl 0157.45803); ibid. 22, 265-278 (1969; Zbl 0174.17402)] stability theory of the class \(I_n\) of isometric mappings from the space \(\mathbb{R}^n\) into itself, \textit{A. P. Kopylov} [Sib. Mat. Zh. 25, No. 2(144), 132-144 (1984; Zbl 0546.30019)] proposed a general approach for the study of the stability problem of Lipschitz mappings, called by him \(\omega\)-stability of the class of this mappings. The author, in the paper under review, establishes the following result in the case of affine mappings: Let \(m,n\in\mathbb{N}\) and \(\mathcal A\) a \(K^*\)-normal class of mappings (cf. A. P. Kopylov's paper quoted above) with \(D{\mathcal A}=0\). Then \(\mathcal A\) is \(\omega\)-stable and there exists a function \(\sigma:[0,+\infty[\to[0,+\infty[\) such that 1) \(\sigma(\varepsilon)\to\sigma(0)=0\) if \(\varepsilon\to 0\) and 2) for each domain \(\Delta\subset\mathbb{R}^n\), star-like with respect to one of its points, for each mappin…
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 923561
zbMATH Keywords: stability
zbMATH Keywords: Lipschitz mappings
zbMATH Keywords: Kopylov normality conditions
zbMATH Keywords: \(\omega\)-saturation
zbMATH Keywords: affine mappings
reviewed by: Petru Caraman
MaRDI profile type: Publication
cites work: Rotation and strain
cites work: On quasi‐isometric mappings, I
cites work: On quasi‐isometric mappings, II
cites work: Stability of isometric mappings
cites work: Q3992482
cites work: Q3994667
cites work: Q4278586
cites work: Q4077267
full work available at URL: https://doi.org/10.1007/bf02112534
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description: scientific article; zbMATH DE number 923561