Let be a compact set in an open set on a Stein manifold of dimension . We denote by the Banach space of all bounded and analytic in functions endowed with the uniform norm and by a compact subset of the space consisted of all restrictions of functions from the unit ball . In 1950ies Kolmogorov posed a problem: does
where is the -entropy of the compact . 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 wih respect to the unit ball in the space . 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 and with good estimates on sublevel sets of extremal plurisubharmonic function for the pair (condenser) . The survey is concluded by a discussion of some open problems.
Soit un sous-ensemble compact d’un ouvert d’une variété de Stein de dimension . On désigne par l’espace de Banach des fonctions analytiques et bornées sur functions muni de la norme de la convergence uniforme et par une partie compacte de l’espace consituée de toutes les restrictions des fonctions de la boule unité . Dans les année 1950, Kolmogorov posa le problème suivant : Existe-t-il une constante telle qu’on ait l’asymptotique
où est la -entropie du compact ? 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 par rapport à la boule unité de l’espace . 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 and avec de bonnes estimations sur les ensembles de sous-niveau de la fonction plurisouharmonique extrémale de la paire (condensateur) (condenser) . On conclut cette présentation avec une discussion des problèmes ouverts.
@article{AFST_2011_6_20_S2_211_0, author = {Zakharyuta, Vyacheslav}, title = {Extendible bases and {Kolmogorov} problem on asymptotics of entropy and widths of some class of analytic functions}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {211--239}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 20}, number = {S2}, year = {2011}, doi = {10.5802/afst.1313}, zbl = {1250.32030}, mrnumber = {2858175}, language = {en}, url = {http://www.numdam.org/articles/10.5802/afst.1313/} }
TY - JOUR AU - Zakharyuta, Vyacheslav TI - Extendible bases and Kolmogorov problem on asymptotics of entropy and widths of some class of analytic functions JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2011 SP - 211 EP - 239 VL - 20 IS - S2 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1313/ DO - 10.5802/afst.1313 LA - en ID - AFST_2011_6_20_S2_211_0 ER -
%0 Journal Article %A Zakharyuta, Vyacheslav %T Extendible bases and Kolmogorov problem on asymptotics of entropy and widths of some class of analytic functions %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2011 %P 211-239 %V 20 %N S2 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1313/ %R 10.5802/afst.1313 %G en %F AFST_2011_6_20_S2_211_0
Zakharyuta, Vyacheslav. 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, Serie 6, Numéro Spécial : Actes du colloque Analyse Complexe et Applications en l’honneur de Nguyen Than Van, Volume 20 (2011) no. S2, pp. 211-239. doi : 10.5802/afst.1313. http://www.numdam.org/articles/10.5802/afst.1313/
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