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Mathematics of the USSR-Sbornik, 1970, Volume 11, Issue 1, Pages 75–88
DOI: https://doi.org/10.1070/SM1970v011n01ABEH002063
(Mi sm3437)
 

This article is cited in 10 scientific papers (total in 10 papers)

Spaces of functions of one variable, analytic in open sets and on compacta

V. P. Zaharyuta
References:
Abstract: $A(K)$ is the space of functions analytic on the compactum $K$ of the extended complex plane $\widehat{\mathbf C}$ with the usual locally convex topology; $\overline A_1=A(\{z:|z|\leqslant1\})$, $\overline A_0=\overline A(\{0\})$.
The following assertions are proved:
1. For the spaces $A(K)$ and $\overline A_1$ to be isomorphic, it is necessary and sufficient that the set $D =\widehat{\mathbf C}\setminus K$ have no more than a finite number of connected components and that the compactum $K$ be regular (i.e. the Dirichlet problem is solvable in $D$ for any continuous function on $\partial D$).
2. For $A(K)$ and $\overline A_0$ to be isomorphic, it is necessary and sufficient that the logarithmic capacity of the compactum $K$ be equal to zero.
3. For $A(K)$ and $\overline A_0\times\overline A_1$ to be isomorphic, it is necessary and sufficient that the compactum $K$ be represented in the form of the sum of two disjoint nonempty compacta, one of which has zero capacity and the other of which is regular and has a complement consisting of no more than a finite number of connected components.
Dual results are obtained for the space $A(D)$, where $D$ is an open set.
Bibliography: 20 titles.
Received: 21.07.1969
Bibliographic databases:
UDC: 517.53+513.881
Language: English
Original paper language: Russian
Citation: V. P. Zaharyuta, “Spaces of functions of one variable, analytic in open sets and on compacta”, Math. USSR-Sb., 11:1 (1970), 75–88
Citation in format AMSBIB
\Bibitem{Zah70}
\by V.~P.~Zaharyuta
\paper Spaces of functions of one variable, analytic in open sets and on compacta
\jour Math. USSR-Sb.
\yr 1970
\vol 11
\issue 1
\pages 75--88
\mathnet{http://mi.mathnet.ru//eng/sm3437}
\crossref{https://doi.org/10.1070/SM1970v011n01ABEH002063}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=438097}
\zmath{https://zbmath.org/?q=an:0193.41202|0216.15602}
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  • https://www.mathnet.ru/eng/sm/v124/i1/p84
  • This publication is cited in the following 10 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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