Application of the real Hardy–Sobolev space on the line for finding the best rational approximations in $L_p$

This work is dedicated to developing methods of the real Hardy–Sobolev space on the line for finding the best rational approximations in the $L_p$ space. The methods considered are based on representing a function of this space as a sum of simple functions and the application of a Cauchy-type integral. Sufficient conditions for a function's membership in the considered space have been obtained and inequalities for assessing the corresponding $\sigma$-norm have been proven. Using the obtained results, exact order estimates of the best rational approximations of certain functions have been found. In particular, from the obtained results, the well-known estimate of the best rational approximations of a function of bounded variation follows.