Extendible bases and Kolmogorov problem on asymptotics of entropy and widths of some class of analytic functions

Vyacheslav Zakharyuta

doi:10.5802/afst.1313

Vyacheslav Zakharyuta ¹

¹ Faculty of Engineering and Natural Sciences, Sabancı University, Orhanlı, 34956 Tuzla/Istanbul, Turkey

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. S2, pp. 211-239.

Résumé
Abstract

Soit $K$ un sous-ensemble compact d’un ouvert $D$ d’une variété de Stein $Ω$ de dimension $n$ . On désigne par $H^{\infty} (D)$ l’espace de Banach des fonctions analytiques et bornées sur $D$ functions muni de la norme de la convergence uniforme et par $A_{K}^{D}$ une partie compacte de l’espace $C (K)$ consituée de toutes les restrictions des fonctions de la boule unité $𝔹_{H^{\infty} (D)}$ . Dans les année 1950, Kolmogorov posa le problème suivant : Existe-t-il une constante $τ$ telle qu’on ait l’asymptotique

ℋ_{ε} (A_{K}^{D}) \sim τ {(ln \frac{1}{ε})}^{n + 1}, ε \to 0,

où $ℋ_{ε} (A_{K}^{D})$ est la $ε$ -entropie du compact $A_{K}^{D}$ ? On donne ici une revue des résultats concernant ce problème et un problème qui lui est relié concernant l’asymptotique stricte des diamètres de Kolmogorov de l’ensemble $A_{K}^{D}$ par rapport à la boule unité de l’espace $C (K)$ . On décrit les progrès réalisés dans l’étude de ces problèmes, commençant par les résultats initiaux des années 1950, en relation étroite avec le problème de l’existence des bases communes pour les espaces $A (K)$ and $A (D)$ avec de bonnes estimations sur les ensembles de sous-niveau de la fonction plurisouharmonique extrémale de la paire (condensateur) (condenser) $(K, D)$ . On conclut cette présentation avec une discussion des problèmes ouverts.

Let $K$ be a compact set in an open set $D$ on a Stein manifold $Ω$ of dimension $n$ . We denote by $H^{\infty} (D)$ the Banach space of all bounded and analytic in $D$ functions endowed with the uniform norm and by $A_{K}^{D}$ a compact subset of the space $C (K)$ consisted of all restrictions of functions from the unit ball $𝔹_{H^{\infty} (D)}$ . In 1950ies Kolmogorov posed a problem: does

ℋ_{ε} (A_{K}^{D}) \sim τ {(ln \frac{1}{ε})}^{n + 1}, ε \to 0,

where $ℋ_{ε} (A_{K}^{D})$ is the $ε$ -entropy of the compact $A_{K}^{D}$ . We give here a survey of results concerned with this problem and a related problem on the strict asymptotics of Kolmogorov diameters of the set $A_{K}^{D}$ wih respect to the unit ball in the space $C (K)$ . We decribe a progress in studying of these problems, beginning with initial results of 1950ies, in the closed connection with the problem on existence of a common basis for the spaces $A (K)$ and $A (D)$ with good estimates on sublevel sets of extremal plurisubharmonic function for the pair (condenser) $(K, D)$ . The survey is concluded by a discussion of some open problems.

MR Zbl

DOI : 10.5802/afst.1313

Affiliations des auteurs :

Vyacheslav Zakharyuta ¹

¹ Faculty of Engineering and Natural Sciences, Sabancı University, Orhanlı, 34956 Tuzla/Istanbul, Turkey

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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Vyacheslav Zakharyuta. Extendible bases and Kolmogorov problem on asymptotics of entropy and widths of some class of analytic functions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. S2, pp. 211-239. doi : 10.5802/afst.1313. https://afst.centre-mersenne.org/articles/10.5802/afst.1313/

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