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June 2011 On asymptotics of entropy of a class of analytic functions
Vyacheslav Zakharyuta
Funct. Approx. Comment. Math. 44(2): 307-315 (June 2011). DOI: 10.7169/facm/1308749134


Let $(K,D)$ be a compact subset of an open set $D$ on a Stein manifold $\Omega$ of dimension $n$, $H^\infty(D)$ the Banach space of all bounded and analytic in $D$ functions endowed with the uniform norm, and $A_{K}^{D}$ be a compact subset in the space of continuous functions $C(K)$ consisted of all restrictions of functions from the unit ball $\mathbb{B}_{H^\infty(D)}$. In 1950s Kolmogorov raised the problem of a strict asymptotics ([K1,K2,KT]) of an entropy of this class of analytic functions: $\mathcal{H}_{\varepsilon}(A_{K}^{D})\sim\tau(\ln\frac{1}{\varepsilon})^{n+1},\varepsilon\rightarrow 0,$ with a constant $\tau $. The main result of this paper, which generalizes and strengthens the Levin's and Tikhomirov's result in [LT], shows that this asymptotics is equivalent to the asymptotics for the widths (Kolmogorov diameters): $\ln d_{k}(A_{K}^{D})\sim -\sigma k^{1/n}, k\rightarrow \infty $, with the constant $\sigma =(\frac{2}{\tau( n+1)})^{1/n}$. This result makes it possible to get a positive solution of the above entropy problem by applying recent results [Z2] on the asymptotics for the widths $d_{k}(A_{K}^{D})$.


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Vyacheslav Zakharyuta. "On asymptotics of entropy of a class of analytic functions." Funct. Approx. Comment. Math. 44 (2) 307 - 315, June 2011.


Published: June 2011
First available in Project Euclid: 22 June 2011

zbMATH: 1218.28010
MathSciNet: MR2841189
Digital Object Identifier: 10.7169/facm/1308749134

Primary: 28D20, 322A07, 47B06
Secondary: 32U20, 32U35

Rights: Copyright © 2011 Adam Mickiewicz University


Vol.44 • No. 2 • June 2011
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