\documentclass[12pt,draft,a4paper]{amsart}
\usepackage{amsmath,amsthm}
\usepackage[T2A]{fontenc}
\usepackage[cp1251]{inputenc}
\usepackage[english]{babel}
\usepackage{amsfonts}
\usepackage{latexsym}
%\usepackage{srctex}
\tolerance 200
\newcommand{\hh}{\widetilde H_{\infty,\beta}}
\newcommand{\hr}{\hh^r}
\renewcommand{\labelenumi}{$\theenumi.$}

\DeclareMathOperator*{\IM}{Im}

\begin{document}
\title{Approximation of analytic functions}

\author{K.~Yu.~Osipenko}

\maketitle


We consider the so-called Hardy--Sobolev classes $\hr$ which are the sets
of all $2\pi$-periodic and analytic in the strip $S_\beta:=\{z:|\IM z|<\beta\}$ functions such that
$$|f^{(r)}(z)|\le1,\quad z\in S_\beta.$$
When $r=0$ these classes are known as the Hardy classes. For these classes
we study the following three themes:
\begin{enumerate}
\item Optimal interpolation.
\item Optimal qudratures.
\item $n$-Widths.
\end{enumerate}

All these problems were studied very intensively for smooth functions,
especially for the Sobolev classes. By the efforts of many mathematicians a
general theory was built for the classes which can be represented as a
convolution with cyclic variation diminishing or totally positive kernels.

This general theory can not be applied to the classes of analytic functions
because as usual they are not represented in such form. Methods which are
usually used for smooth functions are based on the calculation of sign
changes of functions. The number of sign changes is found with the help of
Rolle's theorem. But Rolle's theorem is not valid for complex-valued
functions. It is not even clear what will replace such notion as sign
changes in the complex case.

Nevertheless several results which are very close to the smooth case can be
obtained in the analytic case. Moreover, sometimes a general theory for
smooth and analytic cases can be built. As a consequence of this theory we
determine exact values of $n$-widths both for the Sobolev and
Hardy--Sobolev classes $\hh^{r,\mathbb R}$ which are the sets of all functions
from $\hr$ that are real on the real axis.

\vskip20pt
K.~Yu.~Osipenko

Moscow State University of Aviation Technology

Petrovka 27, Moscow, 103767, Russia

osipenko@mati.msk.su
\end{document}
