Free interpolation sets for Hölder classes

Let $\mathbf D=\{z,|z|<1\}$, let $E$ be a closed subset of $\overline{\mathbf D}$ and let $0<s<1$. Let $A^s$ be the space of functions $f$ analytic in $\mathbf D$ and continuous in $\overline{\mathbf D}$ such that
\begin{equation} |f(z_1)-f(z_2)|\leqslant\operatorname{const}\cdot|z_1-z_2|^s \tag{\ast} \end{equation}
everywhere in $\overline{\mathbf D}$. Let $\Lambda^s(E)$ be the space of functions $f$ continuous on $E$ that satisfy ($\ast$) everywhere on $E$. It is clear that $A^s|_E\subset\Lambda^s(E)$. The set $E$ is said to be $A^s$-interpolating if $A^s|_E=\Lambda^s(E)$.
The article gives necessary and sufficient conditions for a set $E$ to be interpolating (independently of $s$). Similar results are obtained for $s>1$ and for classes of functions with derivatives in $H^p$.
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