Extremal problems for linear functionals on the Tchebycheff spaces

The study of the Tchebycheff spaces (generalizing the space of algebraic polynomials) and extremal problems related to them began one and a half centuries ago. Lately, many facts of the approximation theory were understood and reinterpreted from the point of view of general principles of the theory of extremum and convex duality. This approach not only allowed to prove the previously known results for algebraic polynomials and generalized polynomials in a unified way, but also enabled obtaining new results. In this paper, we work out this direction with a special attention to the optimal recovery problems.