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Mat. Zametki, 1998, Volume 63, Issue 1, Pages 56–61 (Mi mz1247)  

This article is cited in 1 scientific paper (total in 1 paper)

Remarks on the descriptive metric characterization of singular sets of analytic functions

S. V. Kolesnikov

Ivanovo State University

Abstract: This work presents two remarks on the structure of singular boundary sets of functions analytic in the unit disk $D$: $|z|<1$. The first remark concerns the conversion of the Plessner theorem. We prove that three pairwise disjoint subsets $E_1,E_2$, and $E_3$ of the unit circle $\Gamma$: $|z|=1$, $\bigcup_{i=1}^3E_i=\Gamma$ are the sets $I(f)$ of all Plessner points, $F(f)$ of all Fatou points, and $E(f)$ of all exceptional boundary points, respectively, for a function $f$ holomorphic in $D$ if and only if $E_1$ is a $G_\delta$-set and $E_3$ is a $G_{\delta\sigma}$-set of linear measure zero. In the second part of the paper it is shown that for any $G_{\delta\sigma}$-subset $E$ of the unit circle $\Gamma$ with a zero logarithmic capacity there exists a one-sheeted function on $D$ whose angular limits do not exist at the points of $E$ and do exist at all the other points of $\Gamma$.

DOI: https://doi.org/10.4213/mzm1247

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English version:
Mathematical Notes, 1998, 63:1, 50–54

Bibliographic databases:

UDC: 517.514.72
Received: 26.03.1996
Revised: 20.04.1997

Citation: S. V. Kolesnikov, “Remarks on the descriptive metric characterization of singular sets of analytic functions”, Mat. Zametki, 63:1 (1998), 56–61; Math. Notes, 63:1 (1998), 50–54

Citation in format AMSBIB
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\by S.~V.~Kolesnikov
\paper Remarks on the descriptive metric characterization of singular sets of analytic functions
\jour Mat. Zametki
\yr 1998
\vol 63
\issue 1
\pages 56--61
\mathnet{http://mi.mathnet.ru/mz1247}
\crossref{https://doi.org/10.4213/mzm1247}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1631836}
\zmath{https://zbmath.org/?q=an:0917.30019}
\transl
\jour Math. Notes
\yr 1998
\vol 63
\issue 1
\pages 50--54
\crossref{https://doi.org/10.1007/BF02316142}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000075520700006}


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    This publication is cited in the following articles:
    1. Danielyan A.A., Totik V., “A converse to a theorem of Salem and Zygmund”, Bull. Sci. Math., 140:3 (2016), 260–272  crossref  mathscinet  zmath  isi  elib  scopus
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