Parfinovych N. Relative approximations of functional classes

The object of the research is relative approximations and relative widths of Sobolev's classes. Obtaining some new results as to asymptotic and exact values of the best relative approximations and order behaviour of relative widths of Sobolev's classes and some of their generalizations are the aim of the research. Modern methods of theory of functions, functional analysis and approximation theory have been used in this paper. The exact order of relative widths of Sobolev's classes in the space L1 is obtained. The exact order of relative widths of classes of convolutions in the metric Lq has been also obtained. The exact asymptotic of Sobolev's classes best approximations by splines with bounded head derivatives of different orders have been obtained. The conditions under which the best relative approximations of Sbolev's classes by splines coincide with approximations of these classes without restrictions have been obtained. The results are new. They have been pablished in leading scientific editions and they can be used in further investigations of extremal problems of approximation theory.