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Best linear approximations of functions analytically continuable from a given continuum into a given region
V. D. Erokhin
Abstract:
Let
$K$ be a continuum (other than a single point) in the
$z$-plane not disconnecting the plane,
$\mathfrak{G}$ a simply-connected domain containing
$K$. The class
$A_K^{\mathfrak{G}}$ consists of those functions that are analytic in
$\mathfrak{G}$ and satisfy the inequality
$$
|f(z)|\leqslant 1,\quad\mathbf\forall_z\in\mathfrak{G}.
$$
The author proves the following theorem:
$$
H_\varepsilon(A_K^\mathfrak{G})\sim\tau\log_2^2\frac{1}{\varepsilon},\qquad\lim_{n\to\infty}d_n(A_K^\mathfrak{G})]^{\frac{1}{n}}=2^{-\frac{1}{\tau}}.
$$
Here
$H_\varepsilon$ is the
$\varepsilon$-etropy of
$A_K^{\mathfrak{G}}$, and
$d_n$ the
$n$-dimensional linear diameter of
$ A_K^{\mathfrak{G}}$ in the space
$ C(K)$ of all functions continuous on
$K$. The norm on
$ A_K^{\mathfrak{G}}$ is
$$
||f(z)||=\max_{z\in K}|f(z)|.
$$
For the proof a basis is constructed in the space
$\mathscr H(\mathfrak{G})$ of functions holomorphic in
$\mathfrak{G}$; it coincides with the Faber basis if
$\partial\mathfrak{G}$ is a level curve of
$K$. A fundamental part in this construction is played by a lemma which states that the domain
$\mathfrak{G}\setminus K$ can be mapped conformally into a domain
$\mathfrak{G}'\setminus K'$, where
$\partial\mathfrak{G}'$ is a level curve of
$K'$.
In the appendix, which is written by A. L. Levin and V. M. Tikhomirov, a similar theorem is proved (under additional assumptions) for the case when
$\mathfrak{G}$ is multiply-connected and
$K$ may consist of several continua.
UDC:
517.5
MSC: 46A32,
46Gxx,
41A46,
46A45 Received: 08.08.1967