On stability of classes of affine mappings
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 mapping \(f:\Delta\to\mathbb{R}^m\) with \(\Omega(f,{\mathcal G})<\infty\), there exists a mapping \(g:\Delta\to\mathbb{R}^m\) of the class \(\mathcal A\) such that \(\sup_{y\in\Delta}|f(y)-g(y)|\leq\sigma[\Omega(f,{\mathcal A})]\text{diam }\Delta\). In the particular case \(m=n=1\), \(\mathcal A\) is \(\omega\)-stable iff \(D{\mathcal A}=0\). If \(\mathcal A\) is such that \(D{\mathcal A}=\{t\alpha+\beta,t\in J\}\), where \(\alpha,\beta\in L(\mathbb{R}^n,\mathbb{R}^m)\), \(\alpha\neq 0\), \(\text{rank }\alpha\geq 2\) and \(J\) is a compact set in \(\mathbb{R}\), then, \(\mathcal A\) is \(\omega\)-stable. In the particular case \(J=\{t\in\mathbb{R}^n: a\leq t\leq b\}\), \(a,b\in\mathbb{R}\), \(a<b\), then, \(\text{rank }\alpha\geq 2\) iff \(\mathcal A\) is \(\omega\)-stable. Finally, he establishes also a criterium of \(\omega\)-saturation [cf. \textit{A. P. Kopylov}, ``Stability in the \(C\)-norm of classes of mappings'' (1990; Zbl 0772.30023)].
scientific article; zbMATH DE number 923561
scientific article