Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Bernd Carl¹ ; David E. Edmunds²

¹Mathematisches Institut Universität Jena D-07740 Jena, Germany
²School of Mathematical Sciences University of Sussex Brighton BN1 9QH, UK

Studia Mathematica, Tome 159 (2003) no. 3, pp. 391-402

Résumé

For a precompact subset $K$ of a Hilbert space we prove the following inequalities: $$ n^{1/2} c_n(\mathop{\rm cov}\nolimits(K))\le c_K\Big(1+\sum^n_{k=1} k^{-1/2}e_k(K)\Big),\quad\ n\in \mathbb N, $$ and $$ k^{1/2} c_{k+n}(\mathop{\rm cov}\nolimits(K))\le c\bigg[\log^{1/2}(n+1)\varepsilon_n(K)+\sum_{j=n+1}^\infty \frac{\varepsilon_j(K)}{j\log^{1/2}(j+1)}\bigg], $$ $k,n\in\mathbb N$, where $c_n(\mathop{\rm cov}\nolimits(K))$ is the $n$th Gelfand number of the absolutely convex hull of $K$ and $\varepsilon_k(K)$ and $e_k(K)$ denote the $k$th entropy and $k$th 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 $c_n(\mathop{\rm cov}\nolimits(K))$ provided that the entropy numbers $\varepsilon_n(K)$ are slowly decreasing. For example, we get optimal estimates in the non-critical case where $\varepsilon_n(K)\preceq \log^{-\alpha}(n+1)$, $\alpha\not=1/2$, $0\alpha\infty$, as well as in the critical case where $\alpha=1/2$. For $\alpha=1/2$ we show the asymptotically optimal estimate $c_n(\mathop{\rm cov}\nolimits(K))\preceq n^{-1/2}\log(n+1)$, 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 $K=\{t_1,t_2,\ldots\}$, where $\|t_n\|\le \sigma_n$, $ \sigma_n\downarrow 0$. In particular, for $\sigma_n\le \log^{-1/2}(n+1)$, which corresponds to the critical case, we get a better asymptotic behaviour of Gelfand numbers, $c_n(\mathop{\rm cov}\nolimits(K))\preceq n^{-1/2}$.

Détail

DOI : 10.4064/sm159-3-4

Keywords: precompact subset hilbert space prove following inequalities mathop cov nolimits sum quad mathbb mathop cov nolimits bigg log varepsilon sum infty frac varepsilon log bigg mathbb where mathop cov nolimits nth gelfand number absolutely convex hull varepsilon denote kth entropy kth dyadic entropy number respectively inequalities essentially reformulation corresponding inequalities given ckp which yield asymptotically optimal estimates gelfand numbers mathop cov nolimits provided entropy numbers varepsilon slowly decreasing example get optimal estimates non critical where varepsilon preceq log alpha alpha alpha infty critical where alpha alpha asymptotically optimal estimate mathop cov nolimits preceq log which refines corresponding result gao obtained entropy numbers furthermore establish inequalities similar creutzig steinwart crst critical non critical cases finally alternative proof result linde gelfand entropy numbers absolutely convex hull has shape ldots where sigma sigma downarrow particular sigma log which corresponds critical get better asymptotic behaviour gelfand numbers mathop cov nolimits preceq