Functiones et Approximatio Commentarii Mathematici

On asymptotics of entropy of a class of analytic functions

Vyacheslav Zakharyuta

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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 $d_{k}(A_{K}^{D})$.

Article information

Funct. Approx. Comment. Math., Volume 44, Number 2 (2011), 307-315.

First available in Project Euclid: 22 June 2011

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Primary: 28D20: Entropy and other invariants 47B06: Riesz operators; eigenvalue distributions; approximation numbers, s- numbers, Kolmogorov numbers, entropy numbers, etc. of operators 322A07
Secondary: 32U35: Pluricomplex Green functions 32U20: Capacity theory and generalizations

Entropy and widths asymptotics spaces of analytic functions Kolmogorov problem Bedford Taylor capacity of a condenser


Zakharyuta, Vyacheslav. On asymptotics of entropy of a class of analytic functions. Funct. Approx. Comment. Math. 44 (2011), no. 2, 307--315. doi:10.7169/facm/1308749134.

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