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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Author: Joseph W. Jerome
Journal: Proc. Amer. Math. Soc. 33 (1972), 367-372
MSC: Primary 41A65; Secondary 47F05
MathSciNet review: 0296583
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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 $.

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Keywords: n-widths, asymptotic distribution, eigenvalues, elliptic, quadratic inequalities, selfadjoint, coercive, restricted cone condition, Sobolev spaces
Article copyright: © Copyright 1972 American Mathematical Society