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Replacing points by compacta in neural network approximation. (English) Zbl 1075.54013
Dugundji remarks that in Hausdorff spaces, “the compact subsets behave as points do and have the same separation properties”. That is, compacta can replace points in some mathematical assertions. In the paper under review this paradigm of Dugundji is applied to some problems which arise in nonlinear approximation. The author prove statements about realization of distances by pairs of points in metric spaces, about approximative compactness in Cartesian products of metric spaces, about various type of compactness of sums of two sets in linear metric spaces. A typical result is Theorem 5.1: Let \(S\) and \(P\) be nonempty subsets of metric spaces \(X\) and \(Y\), respectively. Suppose that \(P\) is compact. If \(S\) is boundedly compact or approximatively compact, then so is \({S\times P}\).
Applications are given to approximation of \({L_p}\)-functions on the \(d\)-dimensional cube, \({1\leq p<\infty}\), by linear combinations of half-space characteristic functions; i.e., by Heaviside perceptron networks.

MSC:
54E35 Metric spaces, metrizability
54B10 Product spaces in general topology
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
68T05 Learning and adaptive systems in artificial intelligence
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