On \(\varepsilon\)-entropy of Sobolev and Nikolsky classes in uniform metrics on arbitrary compacts (Q1340305)

This paper is devoted to the study of \(\varepsilon\)-entropy of the Nikolsky class \(H^ \alpha_ \infty (I^ s)\) in \(C(K)\), where \(K\) is an arbitrary compact set in \(I^ s : = [0,1]^ s \subset \mathbb{R}^ s\). It was shown by \textit{A. N. Kolmogorov} and \textit{V. M. Tikhomirov} [Usp. Mat. Nauk 14, No. 2(86), 3-86 (1959; Zbl 0133.067)] that for a connected set \(K\) the order of \(\varepsilon\)-entropy is the same as the order of Kolmogorov's \(\varepsilon\)-dimension. This is not the case if \(K\) is not connected. In this paper the exact order is given in terms of two functions characterizing ``density'' and ``discontinuity'' of the compact set \(K\).