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

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 (2011)

  • Volume: 20, Issue: S2, page 211-239
  • ISSN: 0240-2963

Abstract

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Let K be a compact set in an open set D on a Stein manifold Ω of dimension n . We denote by H ∞ 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 ∞ D . In 1950ies Kolmogorov posed a problem: does ℋ ε A K D ∼ τ ln 1 ε n + 1 , ε → 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.

How to cite

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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 20.S2 (2011): 211-239. <http://eudml.org/doc/219695>.

@article{Zakharyuta2011,
abstract = {Let $K$ be a compact set in an open set $D$ on a Stein manifold $\Omega $ of dimension $n$. We denote by $H^\{\infty \}\left( D\right) $ 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\left( K\right) $ consisted of all restrictions of functions from the unit ball $\{\mathbb\{B\}\} _\{H^\{\infty \}\left( D\right) \}$. In 1950ies Kolmogorov posed a problem: does\[ \{\cal H\}\_\{\varepsilon \}\left( A\_\{K\}^\{D\}\right) \sim \tau \left( \ln \frac\{ 1\}\{\varepsilon \}\right) ^\{n+1\text\{ \}\},\;\varepsilon \rightarrow 0, \]where $\{\cal H\}_\{\varepsilon \}\left( A_\{K\}^\{D\}\right) $ is the $\varepsilon $-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\left( K\right) $. 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\left( K\right) $ and $A\left( D\right) $ with good estimates on sublevel sets of extremal plurisubharmonic function for the pair (condenser) $\left( K,D\right) $. The survey is concluded by a discussion of some open problems.},
affiliation = {Faculty of Engineering and Natural Sciences, Sabancı University, Orhanlı, 34956 Tuzla/Istanbul, Turkey},
author = {Zakharyuta, Vyacheslav},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {entropy and widths asymptotics; spaces of analytic functions; Kolmogorov problem},
language = {eng},
month = {4},
number = {S2},
pages = {211-239},
publisher = {Université Paul Sabatier, Toulouse},
title = {Extendible bases and Kolmogorov problem on asymptotics of entropy and widths of some class of analytic functions},
url = {http://eudml.org/doc/219695},
volume = {20},
year = {2011},
}

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
DA - 2011/4//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - S2
SP - 211
EP - 239
AB - Let $K$ be a compact set in an open set $D$ on a Stein manifold $\Omega $ of dimension $n$. We denote by $H^{\infty }\left( D\right) $ 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\left( K\right) $ consisted of all restrictions of functions from the unit ball ${\mathbb{B}} _{H^{\infty }\left( D\right) }$. In 1950ies Kolmogorov posed a problem: does\[ {\cal H}_{\varepsilon }\left( A_{K}^{D}\right) \sim \tau \left( \ln \frac{ 1}{\varepsilon }\right) ^{n+1\text{ }},\;\varepsilon \rightarrow 0, \]where ${\cal H}_{\varepsilon }\left( A_{K}^{D}\right) $ is the $\varepsilon $-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\left( K\right) $. 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\left( K\right) $ and $A\left( D\right) $ with good estimates on sublevel sets of extremal plurisubharmonic function for the pair (condenser) $\left( K,D\right) $. The survey is concluded by a discussion of some open problems.
LA - eng
KW - entropy and widths asymptotics; spaces of analytic functions; Kolmogorov problem
UR - http://eudml.org/doc/219695
ER -

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