The notion of best approximation was introduced into mathematical analysis by the work of P. L. Chebyshev, who in the 1850s considered some of the properties of polynomials with least deviation from a fixed continuous function (see e.g. [2B]). Since then the development of approximation theory has been closely connected with this notion. In contrast to the early investigations which concentrated on the best approximation of individual functions, since the 1930s more effort has been put into the approximation of classes of functions with prescribed differential or difference properties. A variety of extremal problems naturally arises in this field and the solution of these problems led to the concept of exact constants.

The most powerful methods for solving the extremal problems for the best approximation of functional classes are based on the duality relationships in convex analysis. In this chapter we deal mainly with such relationships. The theorems proved in Sections 1.3-1.5 connect different extremal problems. Their solutions are of independent importance, but we shall also use them as a starting point for obtaining exact solutions in particular cases. General results connected with the best approximation of a fixed element from a metric (in particular from a normed) space and the formulation of extremal problems for approximation of a fixed set are given in Sections 1.1-1.2.