Kolmogorov's $(n,\delta)$-widths of spaces of smooth functions

V. E. Maiorov

Sbornik. Mathematics, Tome 79 (1994) no. 2, pp. 265-279

Résumé

Kolmogorov's $(n,\delta)$-widths of the Sobolev spaces $W_2^r$, equipped with a Gaussian probability measure $\mu$, are studied in the metric of $L_q$: $$ d_{n,\delta}(W_2^r,\mu,L_q)=\inf_{G\subset W_2^r}d_n(W_2^r\setminus G,L_q), $$ where $d_n(K, L_q)$ is Kolmogorov's $n$-width of the set $K$ in the space $L_q$, and the infimum is taken over all possible subsets $G\subset W_2^r$ with measure $\mu(G)\le\delta$, $0\le\delta\le1$. The asymptotic equality $$ d_{n,\delta}(W_2^r,\mu,L_q)\asymp n^{-r-\varepsilon}\sqrt{1+\frac1n\ln\frac1\delta} $$ with respect to $n$ and $\delta$ is obtained, where $1\le q\le\infty$ and $\varepsilon>0$ is an arbitrary number depending only on the measure $\mu$.