Relative widths of Sobolev classes in the uniform and integral metrics

Yu. V. Malykhin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 217-223

Résumé

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 $r$.