Asymptotic estimates of the 𝑛-widths in Hilbert space

Abstract

Let Ω ⊂ R m \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 ( Ω ) {L^2}(\Omega ) which is the realization of a uniformly elliptic operator A of order v v with suitable coefficients and principal part a ( x , ξ ) a(x,\xi ) . Let R \mathcal {R} be the ellipsoid { f : ( R f , f ) ≦ 1 } \{ f:(Rf,f) \leqq 1\} . The L 2 {L^2} n-widths d n ( R ) {d_n}(\mathcal {R}) satisfy d n ( R ) ∼ ( c / n ) v / 2 m {d_n}(\mathcal {R}) \sim {(c/n)^{v/2m}} where c = ∫ Ω ( ∫ 0 > a ( x , ξ ) > 1 d ξ ) d x c = \smallint _\Omega {(\smallint _{0 > a(x,\xi ) > 1} {d\xi )\;dx} } . If B ( u , v ) B(u,v) is a nonnegative Hermitian coercive form over a subspace V \mathcal {V} of the Sobolev space W k , 2 ( Ω ) {W^{k,2}}(\Omega ) , then the n-widths of B = { f ∈ V : B ( f , f ) ≦ 1 } \mathcal {B} = \{ f \in \mathcal {V}:B(f,f) \leqq 1\} satisfy, 0 > ( c ′ ) k / m ≦ lim inf d n ( B ) n k / m ≦ lim sup d n ( B ) n k / m ≦ ( c ) k / m 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 c’ = c = c where c is defined in terms of an elliptic operator of order 2k. The n-widths of B \mathcal {B} in W j , 2 ( Ω ) , 0 ≦ j ≦ k {W^{j,2}}(\Omega ),0 \leqq j \leqq k , are of order O ( n − ( k − j ) / m ) , n → ∞ O({n^{ - (k - j)/m}}),n \to \infty .