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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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