On linear widths of Sobolev classes and chains of extremal subspaces

The linear and trigonometric $n$-diameters of the class $\widetilde W^r_p$ in $L_q$ are calculated in this paper.
For the linear diameter $\lambda_n$ it is proved that, when $p<2<q$ and $r>\frac1p+\frac12$,
$$ \lambda_n(\widetilde W^r_p,L_q)\asymp\begin{cases}n^{-r+\frac1p-\frac12},&\frac1p+\frac1q\leqslant1,\\n^{-r+\frac12-\frac1q},&\frac1p+\frac1q>1.\end{cases} $$

This formula, together with the known results for other $(p,q)$, finishes the solution of the problem of asymptotic computation of the linear diameters for the Sobolev classes in the one-dimensional periodic case when $r>\frac1p+\frac12$.
Bibliography: 28 titles.