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Kolmogorov's $(n,\delta)$-widths of spaces of smooth functions
V. E. Maiorov
Abstract:
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
$\mu$.
UDC:
517.5
MSC: Primary
41A46,
46E35; Secondary
28C20 Received: 16.04.1992
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