Free interpolation sets for H\"older classes

E. M. Dyn'kin

Sbornik. Mathematics, Tome 37 (1980) no. 1, pp. 97-117

Résumé

Let $\mathbf D=\{z,|z|1\}$, let $E$ be a closed subset of $\overline{\mathbf D}$ and let $0$. 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 $H^p$. Bibliography: 18 titles.