Average dimension and \(\nu\)-widths of classes of functions on the whole line (Q1190533)

The notion of \(\varphi\)-average dimension of a linear subspace (generally speaking of infinite-dimension) in a normed linear space of functions on \(\mathbb{R}\) is introduced. On this basis the Kolmogorov \(\varphi\)-average \(\nu\)-width, the linear \(\varphi\)-average \(\nu\)-width and the Bernstein \(\varphi\)-average \(\nu\)-width of a subset in a normed linear space are defined. Exact expressions of these \(\nu\)-widths for the Sobolev class \(W^ r_ p(\mathbb{R})\) in \(L_ p(\mathbb{R})\) are obtained where \(1\leq p\leq\infty\), \(r\) is a positive integer and \(\nu>0\) any real number.