Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Bernd 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 -