Asymptotic estimates of the «-widths in hilbert space

Joseph W. Jerome

doi:10.1090/S0002-9939-1972-0296583-8

Asymptotic estimates of the «-widths in hilbert space

Joseph W. Jerome^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

Let fÃ-eRm be a bounded open set satisfying the restricted cone property and let R be a nonnegative selfadjoint operator on L2(fÃ) which is the realization of a uniformly elliptic operator A of order v with suitable coefficients and principal part a(x, (). Let St be the ellipsoid (f:(Rf, f)\). The Z.2 â-widths d(Ã¢r) satisfy ti,â(fM* where' c = J'â£2(Joa(I.f)i di) dx. If B(u, v) is a nonnegative Hermitian coercive form over a subspace f of the Sobolev space W:2(Q.), then the n-widths of.:#= (/e f:B(f, f)Sl] satisfy, 0(cT'"^h'm inf â â€ž(Ã?)â*'"^ lim sup dn(J8)n::lmS(c")klm. In some cases c'=c=c" where c is defined in terms of an elliptic operator of order 2k. The â-widths of %i n WJ-2(fÃ), Q^jSk, are of order CKâ*-""â), noe.

Original language	English (US)
Pages (from-to)	367-372
Number of pages	6
Journal	Proceedings of the American Mathematical Society
Volume	33
Issue number	2
DOIs	https://doi.org/10.1090/S0002-9939-1972-0296583-8
State	Published - Jun 1972
Externally published	Yes

Keywords

Asymptotic distribution
Eigenvalues
Elliptic
N-widths

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/S0002-9939-1972-0296583-8

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title = "Asymptotic estimates of the «-widths in hilbert space",

abstract = "Let f{\~A}-eRm be a bounded open set satisfying the restricted cone property and let R be a nonnegative selfadjoint operator on L2(f{\~A}) which is the realization of a uniformly elliptic operator A of order v with suitable coefficients and principal part a(x, (). Let St be the ellipsoid (f:(Rf, f)\). The Z.2 {\^a}-widths d({\~A}¢r) satisfy ti,{\^a}(fM* where' c = J'{\^a}£2(Joa(I.f)i di) dx. If B(u, v) is a nonnegative Hermitian coercive form over a subspace f of the Sobolev space W:2(Q.), then the n-widths of.:#= (/e f:B(f, f)Sl] satisfy, 0(cT'{"}^h'm inf {\^a} {\^a}€{\v z}({\~A}?){\^a}*'{"}^ lim sup dn(J8)n::lmS(c{"})klm. In some cases c'=c=c{"} where c is defined in terms of an elliptic operator of order 2k. The {\^a}-widths of %i n WJ-2(f{\~A}), Q^jSk, are of order CK{\^a}*-{"}{"}{\^a}), noe.",

keywords = "Asymptotic distribution, Eigenvalues, Elliptic, N-widths",

author = "Jerome, {Joseph W.}",

year = "1972",

month = jun,

doi = "10.1090/S0002-9939-1972-0296583-8",

language = "English (US)",

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PY - 1972/6

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N2 - Let fÃ-eRm be a bounded open set satisfying the restricted cone property and let R be a nonnegative selfadjoint operator on L2(fÃ) which is the realization of a uniformly elliptic operator A of order v with suitable coefficients and principal part a(x, (). Let St be the ellipsoid (f:(Rf, f)\). The Z.2 â-widths d(Ã¢r) satisfy ti,â(fM* where' c = J'â£2(Joa(I.f)i di) dx. If B(u, v) is a nonnegative Hermitian coercive form over a subspace f of the Sobolev space W:2(Q.), then the n-widths of.:#= (/e f:B(f, f)Sl] satisfy, 0(cT'"^h'm inf â â€ž(Ã?)â*'"^ lim sup dn(J8)n::lmS(c")klm. In some cases c'=c=c" where c is defined in terms of an elliptic operator of order 2k. The â-widths of %i n WJ-2(fÃ), Q^jSk, are of order CKâ*-""â), noe.

AB - Let fÃ-eRm be a bounded open set satisfying the restricted cone property and let R be a nonnegative selfadjoint operator on L2(fÃ) which is the realization of a uniformly elliptic operator A of order v with suitable coefficients and principal part a(x, (). Let St be the ellipsoid (f:(Rf, f)\). The Z.2 â-widths d(Ã¢r) satisfy ti,â(fM* where' c = J'â£2(Joa(I.f)i di) dx. If B(u, v) is a nonnegative Hermitian coercive form over a subspace f of the Sobolev space W:2(Q.), then the n-widths of.:#= (/e f:B(f, f)Sl] satisfy, 0(cT'"^h'm inf â â€ž(Ã?)â*'"^ lim sup dn(J8)n::lmS(c")klm. In some cases c'=c=c" where c is defined in terms of an elliptic operator of order 2k. The â-widths of %i n WJ-2(fÃ), Q^jSk, are of order CKâ*-""â), noe.

KW - Asymptotic distribution

KW - Eigenvalues

KW - Elliptic

KW - N-widths

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Asymptotic estimates of the «-widths in hilbert space

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