Average widths of Sobolev classes on \(\mathbb{R}^ n\)
For \(p= (p_ 1,\dots, p_ n)\), \(1\leq p_ i\leq \infty\), \(i=1,\dots, n\), denote by \(L^ p (\mathbb{R}^ n)\) the Banach space of Lebesgue measurable functions on \(\mathbb{R}^ n\), with a mixed norm in the sense of \textit{A. Benedèk} and \textit{R. Panzone} [Duke Math. J. 28, 301-314 (1961; Zbl 0107.089)]. For a positive nondecreasing function \(\varphi\) on \((0,\infty)\), such that \(\varphi(a)\to \infty\) as \(a\to \infty\), the author defines the \(\varphi\)-average dimension of a subspace of \(L_ p (\mathbb{R}^ n)\) and, for \(\nu\geq 0\), the corresponding Kolmogorov \(\varphi\)-average \(\nu\)-width of a centrally symmetric subset of \(L_ p (\mathbb{R}^ n)\). In the case of the Sobolev class \(W_ p^ r (\mathbb{R}^ n)\) this width is finite if and only if \(\lim_{a\to\infty} (a^ n/ \varphi(a))<\infty\). Its asymptotic behaviour for \(a\to\infty\) is determined. In the case \(\varphi(a)= (2a)^ n\) the exact value of the average \(\nu\)-width of the unit ball of the subspace \(H_ 2^ \alpha (\mathbb{R}^ n)\), \(\alpha>0\), of \(L_ 2(\mathbb{R}^ n)\) is calculated.
scientific article; zbMATH DE number 502998