On the best linear approximation of holomorphic functions

Let $\Omega$ be an open subset of the complex plane $\mathbb C$ and let $E$ be a compact subset of $~\Omega$. The present survey is concerned with linear $n$-widths for the class $H^\infty(\Omega)$ in the space $C(E)$ and some problems on the best linear approximation of classes of Hardy–Sobolev-type in $L^p$-spaces. It is known that the partial sums of the Faber series give the classical method for approximation of functions $f\in H^\infty(\Omega)$ in the metric of $C(E)$ when $E$ is a bounded continuum with simply connected complement and $\Omega$ is a canonical neighborhood of $E$. Generalizations of the Faber series are defined for the case where $\Omega$ is a multiply connected domain or a disjoint union of several such domains, while $E$ can be split into a finite number of continua. The exact values of $n$-widths and asymptotic formulas for the $\varepsilon$-entropy of classes of holomorphic functions with bounded fractional derivatives in domains of tube type are presented. Also, some results about Faber's approximations in connection with their applications in numerical analysis are mentioned.