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Stability of classes of affine mappings. (Russian, English) Zbl 0989.30019
Sib. Mat. Zh. 42, No. 6, 1259-1277 (2001); translation in Sib. Math. J. 42, No. 6, 1047-1061 (2001).
In the article, the following problem is studied: Let \(G\) be a compact set in the space of linear mappings \(L(\mathbb R^n,\mathbb R^m)\). Assume that a mapping \(f\:B(0,1)\to\mathbb R^m\) on the unit ball \(B(0,1)\subset\mathbb R^n\) is approximated by mappings in \(G\) on an infinitesimal neighborhood of every point \(x \in B(0,1)\). Is it possible to conclude that, on the entire ball \(B(0,1)\), the mapping \(f\) is close, in the \(C\)-norm, to an affine mapping with linear part in \(G\)?
If the answer is positive, then the class of affine mappings with linear parts in \(G\) (denoted by \({\mathcal U}(G)\)) is said to be stable. The stability is treated in the framework of the concept of \(\omega\)-stability (or strong stability) suggested by A. P. Kopylov in [Sib. Math. J. 25, 274-283 (1984); translation from Sib. Mat. Zh. 25, No. 2(144), 132-144 (1984; Zbl 0546.30019)]. A special attention is paid to the notion of \(\omega^{\circ}\)-stability (or weak stability).
The authors expose a collection of theorems about stability of classes of affine mappings, and the first result obtained (Theorem 1) guarantees that if for every compact set \(G\subset L(\mathbb R^n,\mathbb R^m)\) there exists a representation of the form \[ G = \bigcap_{\alpha\in A}\bigcup_{i = 1}^{k_{\alpha}}G_i^{\alpha}, \quad G_i^{\alpha}\cap G\cap G_j^{\alpha} = \varnothing, \quad \alpha\in A,\;i \neq j, \;i,j \in \{1,\dots,k_{\alpha}\}, \] where \(G_i^{\alpha}\) are quasiconvex sets and, in addition, the following condition holds: \[ \forall a,b \in G\;(a\neq b\;\Rightarrow\;\exists\alpha\in A\;\exists i\neq j \in \{1,\dots,k_{\alpha}\},\;\alpha \in G_i^{\alpha} \text{and } b \in G_j^{\alpha}), \] then the class \({\mathcal U}(G)\) is \(\omega\)-stable. This theorem implies that the class \({\mathcal U}(G)\) is \(\omega\)-stable if the compact set \(G\) is totally disconnected and the above-mentioned representation for \(G\) exists for convex sets \(G_i^{\alpha}\). The result is complemented by the assertion (Theorem 2) that totally disconnectedness of \(G\) is sufficient for \(\omega\)-stability for the class in the case when \(m = 1\). In Theorem 3, for \(n = 1\), the authors expose both necessary and sufficient conditions for \(\omega\)-stability of the class \({\mathcal U}(G)\). Next, in Theorem 4 the result obtained by A. A. Egorov in [Sib. Math. J. 36, No. 5, 930-942 (1995); translation from Sib. Mat. Zh. 36, No. 5, 1081-1095 (1995; Zbl 0865.30025)] is revised, wherein \(\omega\)-stability was stated of the class \({\mathcal U}(G)\) for an arbitrary totally disconnected set \(G\) and only \(\omega^{\circ}\)-stability was proven. The authors present an exact statement of this result and prove that, for \(n = 1\), totally disconnectedness of the compact set \(G\) is also necessary for \(\omega^{\circ}\)-stability of \({\mathcal U}(G)\).
Finally, the authors note that, for the classes of mappings considered in Theorems 1, 2, and 4, the stability estimates hold in the \(W^1_{\infty}\)-norm.
MSC:
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
51N10 Affine analytic geometry
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