Tchebycheff rational approximation in the disk, on the circle, and on a closed interval

Suppose that the function $f$ is analytic in the disk $\{z:|z|<1\}$ and continuous in its closure. Let $R_n(f)$ denote the best uniform approximation of $f$ by rational functions of degree at most $n$. In 1965 Dolzhenko established that if $\sum R_n(f)<\infty$ then $f'$ belongs to the Hardy space $H_1$. The following converse of this result is obtained here: if $f'\in H_1$, then $R_n(f)=O(1/n)$. In combination with results of Peller, Semmes, and the author, this estimate yields, in particular, a description of the set of functions $f$ with $\bigl[\sum(2^{k\alpha }R_{2^k}(f))^q\bigr]^{1/q}<\infty$, where $\alpha>1$ and $0<q\le\infty$.
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