Kolmogorov Widths of Classes of Analytic Functions: Recent Results and Open Problems

This article is a brief survey of results on Kolmogorov widths of classes of analytic functions in one and several complex variables. The main results on Kolmogorov’s problem of asymptotic estimates of widths in spaces of bounded analytic functions are presented. The widths of Hardy–Sobolev classes on various domains are also discussed. Several examples are given where optimal widths estimates are obtained using suitable modifications of the classical Faber approximations. The bibliography includes recent papers on polynomial approximations and widths of classes of functions on manifolds. Some open problems are formulated.