Application of the real Hardy-Sobolev space on the line for finding the best rational approximations in $L_p$
Abstract
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.
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Review
For citations:
Mardvilko T.S. Application of the real Hardy-Sobolev space on the line for finding the best rational approximations in $L_p$. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2024;32(2):31-42. (In Russ.)
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