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 Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 2, Pages 325–333 (Mi izv1804)

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

K. Val'des Kastro, M. S. Mel'nikov

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 $R(X)$, $ż_m\}_{m=1}^\infty\subset X\setminus P$, 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 $ż_m\}_{m=1}^\infty$ 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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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:2, 308–316

Bibliographic databases:

UDC: 517.5
MSC: Primary 30A80, 30A98; Secondary 30A44

Citation: K. Val'des Kastro, M. S. Mel'nikov, “On interpolation sets for the algebra $R(X)$”, Izv. Akad. Nauk SSSR Ser. Mat., 41:2 (1977), 325–333; Math. USSR-Izv., 11:2 (1977), 308–316

Citation in format AMSBIB
\Bibitem{ValMel77}
\by K.~Val'des Kastro, M.~S.~Mel'nikov
\paper On interpolation sets for the algebra~$R(X)$
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1977
\vol 41
\issue 2
\pages 325--333
\mathnet{http://mi.mathnet.ru/izv1804}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=499175}
\zmath{https://zbmath.org/?q=an:0357.46057|0378.46046}
\transl
\jour Math. USSR-Izv.
\yr 1977
\vol 11
\issue 2
\pages 308--316
\crossref{https://doi.org/10.1070/IM1977v011n02ABEH001716}

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This publication is cited in the following articles:
1. M. S. Mel'nikov, “A criterion for points to belong to the same Gleason part of the algebra $R(X)$”, Math. USSR-Izv., 11:5 (1977), 1109–1117
•  Number of views: This page: 155 Full text: 47 References: 28 First page: 1