On interpolation sets for the algebra~$R(X)$

K. Val'des Kastro ; M. S. Mel'nikov

Izvestiya. Mathematics , Tome 11 (1977) no. 2, pp. 308-316

Résumé

In this paper a connection is established between the behavior of the series remainder $$ R_n(z_m)=\sum_{k=n}^\infty2^k\gamma(A_k(z_m)\setminus X) $$ (where $A_k(z_m)$ is the annulus $\{1/2^{k+1}|z-z_m|1/2^k\}$, and $\gamma$ is analytic capacity) and the Gleason distance $d(z_m,z_0)$ in the algebra $R(X)$, as $z_m\to z_0$. It is proved that if the compact set $X\subset\mathbf C$, $P$ is the set of all peak points of $R(X)$, $\{z_m\}_{m=1}^\infty\subset X\setminus P$, and $z_m\to z_0$ as $m\to\infty$, then in order that $d(z_m,z_0)\to0$ as $m\to\infty$, it is necessary and sufficient that $R_n(z)\to0$ uniformly on the set $\{z_m\}_{m=1}^\infty$ as $n\to\infty$. This result is applied in the study of interpolation sets of the algebra $R(X)$. Bibliography: 10 titles.