On extremal problems in the theory of best approximation

This article is of the nature of a survey. It sets out the basic steps of the investigations into the exact solution of extremal problems in the theory of approximation for classes of periodic functions in their historical aspect (best approximation by trigonometric polynomials, diameters, approximation of one class of functions by another, etc.) Special attention is given to explaining the principles characterizing the various approaches to the solution of the problems. A method worked out by the author and connected with the application of a special operator that is defined by means of rearrangements is expounded in greater detail. This method enables us to obtain an exact solution of certain extremal problems for the classes $W^rH^\omega$.