Classes of analytic functions determined by best rational approximations in $H_p$

Let $R_n(f,H_p)$ be the best approximation to the function $f$ in the Hardy space $H_p$ by rational functions of degree at most $n-1$. It is shown that, for example, $f\in H_p$ ($1<p<\infty$) satisfies the condition $\sum_{k=0}^\infty(2^{k\alpha}R_{2^k}(f,H_p))^\sigma<\infty$ ($\alpha>0$, $\sigma=(\alpha+p^{-1})^{-1}$) if and only if $f$ belongs to the Hardy–Besov space $B_\sigma^\alpha$. Rational approximation is also considered in $H_p$ ($p\leqslant1$) and $H_\infty$. Some applications of the results are given.
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