Geodesic
Application of the real Hardy--Sobolev space on the line for finding the best rational approximations in $L_p$
Trudy Instituta matematiki, Tome 32 (2024) no. 2, pp. 31-42

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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.
Keywords: Hardy space, Sobolev space, Hardy–Sobolev space, rational approximation, $L_p$-approximations, functions of bounded variation. 
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     author = {T. S. Mardvilko},
     title = {Application of the real {Hardy--Sobolev} space on the line for finding the best rational approximations in $L_p$},
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     url = {http://geodesic.mathdoc.fr/item/TIMB_2024_32_2_a2/}
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T. S. Mardvilko. Application of the real Hardy--Sobolev space on the line for finding the best rational approximations in $L_p$. Trudy Instituta matematiki, Tome 32 (2024) no. 2, pp. 31-42. http://geodesic.mathdoc.fr/item/TIMB_2024_32_2_a2/