Skorokhodov D. Inequalities of Kolmogorov type for functions in one variable and their applications

The objects of the thesis are linear and differential operators, classes of multiply monotone and absolutely monotone functions, class S. The purpose of the thesis is to obtain sharp inequalities between derivatives of functions belonging to special functional classes, to solve the problem of approximation of unbounded operators by linear operators that are bounded on these classes, and to solve the Kolmogorov problem for three numbers. The methods used include general methods of solving extremal problems in approximation theory, methods of proving inequalities of Kolmogorov type, methods of estimating the best approximation of unbounded operators by linear bounded ones, general facts from functional analysis, function theory and the theory of vector fields in the plane. The thesis is devoted to exact inequalities between derivatives of functions belonging to special functional classes, problems of finding the quantity of the best approximation of differential operators by linear operators that are bounded on these classes, finding the necessary and sufficient conditions in the Kolmogorov problem for three numbers. New exact inequalities between derivatives and fractional derivatives of functions which are multiply monotone or absolutely monotone on the non-positive half-line or belong to the class S are found. The problem of the best approximation of differential operators by linear operators bounded on the above mentioned functional classes is solved. The Kolmogorov problem for three numbers on above mentioned functional classes and on the class of absolutely monotone on a bounded interval functions is also solved. It is shown that the generalized Kolmogorov problem for three numbers that consists in finding necessary and sufficient conditions for three positive numbers to be the norms of some element in normed space and its images with respect of two linear transformations is equivalent to establishing of abstract versions of Kolmogorov type inequalities between these transformations. The results obtained in the thesis can be used to study the properties of differential operators on special functional classes, to solve Kolmogorov problem for three numbers, and to calculate derivatives numerically.