Relative widths of Sobolev classes in the uniform and integral metrics

Let $W^r_p$ be the Sobolev class consisting of $2\pi$-periodic functions $f$ such that $\|f^{(r)}\|_p\le1$. We consider the relative widths $d_n(W^r_p,MW^r_p,L_p)$, which characterize the best approximation of the class $W^r_p$ in the space $L_p$ by linear subspaces for which (in contrast to Kolmogorov widths) it is additionally required that the approximating functions $g$ should lie in $MW^r_p$, i.e., $\|g^{(r)}\|_p\le M$. We establish estimates for the relative widths in the cases of $p=1$ and $p=\infty$; it follows from these estimates that for almost optimal (with error at most $Cn^{-r}$, where $C$ is an absolute constant) approximations of the class $W^r_p$ by linear $2n$-dimensional spaces, the norms of the $r$th derivatives of some approximating functions are not less than $c\ln\min(n,r)$ for large $n$ and $r$.