Application of the real Hardy-Sobolev space on the line to study the order of uniform rational approximations of functions

The real space of Hardy-Sobolev on a straight line is considered and some sufficient conditions for belonging to functions to this space are described. Estimates of the norm of functions from this space are also obtained. Various examples of functions from the Hardy-Sobolev space are given and the order of their best uniform rational approximations are investigated. Estimates of the best rational approximations for even and odd continuations of functions with monotonous derivatives are obtained. The order of the best rational approximations of the even and odd continuations of functions in the general case have also been studied. Estimates are given both considering the continuity module and without it. The obtained results are also used to study the best rational approximations of functions with a kink, introduced by A. A. Gonchar.