Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Bernd Carl ; David E. Edmunds

Studia Mathematica, Tome 157 (2003), p. 391-402 / Harvested from The Polish Digital Mathematics Library

Résumé

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, $cₙ(cov(K)) ⪯ n^{-1/2}$.

Détail

Publié le : 2003-01-01

Zbl 1054.41018

EUDML-ID : urn:eudml:doc:285355