Best linear approximations of functions analytically continuable from a~given continuum into a~given region

V. D. Erokhin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 23 (1968) no. 1, pp. 93-135

Résumé

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.