Mean dimension, widths, and optimal recovery of Sobolev classes of functions on the line

The concept of mean dimension is introduced for a broad class of subspaces of $L_p(\mathbf R)$, and analogues of the Kolmogorov widths, Bernstein widths, Gel'fand widths, and linear widths are defined. The precise values of these quantities are computed for Sobolev classes of functions on $\mathbf R$ in compatible metrics, and the corresponding extremal spaces and operators are described. A closely related problem of optimal recovery of functions in Sobolev classes is also studied.