# Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Studia Mathematica (2003)

- Volume: 159, Issue: 3, page 391-402
- ISSN: 0039-3223

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topBernd Carl, and David E. Edmunds. "Gelfand numbers and metric entropy of convex hulls in Hilbert spaces." Studia Mathematica 159.3 (2003): 391-402. <http://eudml.org/doc/285355>.

@article{BerndCarl2003,

abstract = {For a precompact subset K of a Hilbert space we prove the following inequalities:
$n^\{1/2\} cₙ(cov(K)) ≤ c_\{K\}(1 + ∑^\{ⁿ\}_\{k=1\} k^\{-1/2\} e_k(K))$, n ∈ ℕ,
and
$k^\{1/2\} c_\{k+n\}(cov(K)) ≤ c[log^\{1/2\}(n+1)εₙ(K) + ∑_\{j=n+1\}^\{∞\} ε_j(K)/(j log^\{1/2\}(j+1))]$,
k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and $ε_k(K)$ and $e_k(K)$ denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K) are slowly decreasing. For example, we get optimal estimates in the non-critical case where $εₙ(K) ⪯ log^\{-α\}(n + 1)$, α ≠ 1/2, 0 < α < ∞, as well as in the critical case where α = 1/2. For α = 1/2 we show the asymptotically optimal estimate $cₙ(cov(K)) ⪯ n^\{-1/2\} log(n + 1)$, which refines the corresponding result of Gao [Ga] obtained for entropy numbers. Furthermore, we establish inequalities similar to that of Creutzig and Steinwart [CrSt] in the critical as well as non-critical cases. Finally, we give an alternative proof of a result by Li and Linde [LL] for Gelfand and entropy numbers of the absolutely convex hull of K when K has the shape K = t₁,t₂,..., where ||tₙ|| ≤ σₙ, σₙ↓ 0. In particular, for $σₙ ≤ log^\{-1/2\}(n + 1)$, which corresponds to the critical case, we get a better asymptotic behaviour of Gelfand numbers, $cₙ(cov(K)) ⪯ n^\{-1/2\}$.},

author = {Bernd Carl, David E. Edmunds},

journal = {Studia Mathematica},

keywords = {metric entropy; Gelfand number; convex sets},

language = {eng},

number = {3},

pages = {391-402},

title = {Gelfand numbers and metric entropy of convex hulls in Hilbert spaces},

url = {http://eudml.org/doc/285355},

volume = {159},

year = {2003},

}

TY - JOUR

AU - Bernd Carl

AU - David E. Edmunds

TI - Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

JO - Studia Mathematica

PY - 2003

VL - 159

IS - 3

SP - 391

EP - 402

AB - For a precompact subset K of a Hilbert space we prove the following inequalities:
$n^{1/2} cₙ(cov(K)) ≤ c_{K}(1 + ∑^{ⁿ}_{k=1} k^{-1/2} e_k(K))$, n ∈ ℕ,
and
$k^{1/2} c_{k+n}(cov(K)) ≤ c[log^{1/2}(n+1)εₙ(K) + ∑_{j=n+1}^{∞} ε_j(K)/(j log^{1/2}(j+1))]$,
k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and $ε_k(K)$ and $e_k(K)$ denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K) are slowly decreasing. For example, we get optimal estimates in the non-critical case where $εₙ(K) ⪯ log^{-α}(n + 1)$, α ≠ 1/2, 0 < α < ∞, as well as in the critical case where α = 1/2. For α = 1/2 we show the asymptotically optimal estimate $cₙ(cov(K)) ⪯ n^{-1/2} log(n + 1)$, which refines the corresponding result of Gao [Ga] obtained for entropy numbers. Furthermore, we establish inequalities similar to that of Creutzig and Steinwart [CrSt] in the critical as well as non-critical cases. Finally, we give an alternative proof of a result by Li and Linde [LL] for Gelfand and entropy numbers of the absolutely convex hull of K when K has the shape K = t₁,t₂,..., where ||tₙ|| ≤ σₙ, σₙ↓ 0. In particular, for $σₙ ≤ log^{-1/2}(n + 1)$, which corresponds to the critical case, we get a better asymptotic behaviour of Gelfand numbers, $cₙ(cov(K)) ⪯ n^{-1/2}$.

LA - eng

KW - metric entropy; Gelfand number; convex sets

UR - http://eudml.org/doc/285355

ER -

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