Asymptotic estimates of the $n$-widths in Hilbert space

Abstract:

Let $\Omega \subset {R^m}$ be a bounded open set satisfying the restricted cone property and let R be a nonnegative selfadjoint operator on ${L^2}(\Omega )$ which is the realization of a uniformly elliptic operator A of order $v$ with suitable coefficients and principal part $a(x,\xi )$. Let $\mathcal {R}$ be the ellipsoid $\{ f:(Rf,f) \leqq 1\}$. The ${L^2}$ n-widths ${d_n}(\mathcal {R})$ satisfy ${d_n}(\mathcal {R}) \sim {(c/n)^{v/2m}}$ where $c = \smallint _\Omega {(\smallint _{0 < a(x,\xi ) < 1} {d\xi )\;dx} }$. If $B(u,v)$ is a nonnegative Hermitian coercive form over a subspace $\mathcal {V}$ of the Sobolev space ${W^{k,2}}(\Omega )$, then the n-widths of $\mathcal {B} = \{ f \in \mathcal {V}:B(f,f) \leqq 1\}$ satisfy, $0 < {(c’)^{k/m}} \leqq \lim \inf {d_n}(\mathcal {B}){n^{k/m}} \leqq \lim \sup {d_n}(\mathcal {B}){n^{k/m}} \leqq {(c'')^{k/m}}$ . In some cases $c’ = c = c''$ where c is defined in terms of an elliptic operator of order 2k. The n-widths of $\mathcal {B}$ in ${W^{j,2}}(\Omega ),0 \leqq j \leqq k$, are of order $O({n^{ - (k - j)/m}}),n \to \infty$.