Abstract:
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.
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