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Mat. Sb. (N.S.), 1984, Volume 125(167), Number 2(10), Pages 269–279 (Mi msb2082)  

This article is cited in 5 scientific papers (total in 5 papers)

A criterion for rapid rational approximation in $\mathbf C^n$

A. S. Sadullaev


Abstract: This article gives a necessary and sufficient condition for a function which is holomorphic in a neighborhood of zero to belong to the class $R^0$. This criterion, which is formulated in terms of the Taylor coefficients of the function, is then applied to give a description of the singular set of holomorphic functions of several variables which admit rapid rational approximation relative to Lebesgue measure (i.e., which belongs to the class $R^0$). In particular,
Theorem. If $\mathscr O(D)\subset R^0$, then the complement $\mathbf C^n\setminus\widehat D$ of the envelope of holomorphy $D$ is a pluripolar set.
This theorem together with a well-known result of A. A. Gonchar gives a complete description of the domains for which $\mathscr O(D)\subset R^0$: this property is satisfied if and only if $\mathbf C^n\setminus\widehat D$ is a pluripolar set.
Bibliography: 11 titles.

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English version:
Mathematics of the USSR-Sbornik, 1986, 53:1, 271–281

Bibliographic databases:

UDC: 517.55
MSC: Primary 32E30; Secondary 31B15
Received: 13.10.1983

Citation: A. S. Sadullaev, “A criterion for rapid rational approximation in $\mathbf C^n$”, Mat. Sb. (N.S.), 125(167):2(10) (1984), 269–279; Math. USSR-Sb., 53:1 (1986), 271–281

Citation in format AMSBIB
\Bibitem{Sad84}
\by A.~S.~Sadullaev
\paper A~criterion for rapid rational approximation in~$\mathbf C^n$
\jour Mat. Sb. (N.S.)
\yr 1984
\vol 125(167)
\issue 2(10)
\pages 269--279
\mathnet{http://mi.mathnet.ru/msb2082}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=764481}
\zmath{https://zbmath.org/?q=an:0592.32013}
\transl
\jour Math. USSR-Sb.
\yr 1986
\vol 53
\issue 1
\pages 271--281
\crossref{https://doi.org/10.1070/SM1986v053n01ABEH002920}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. S. Sadullaev, E. M. Chirka, “On continuation of functions with polar singularities”, Math. USSR-Sb., 60:2 (1988), 377–384  mathnet  crossref  mathscinet  zmath
    2. Bloom T., “On the Convergence in Capacity of Rational Approximants”, Constr. Approx., 17:1 (2001), 91–102  crossref  mathscinet  zmath  isi
    3. Edigarian A., Wiegerinck J., “The Pluripolar Hull of the Graph of a Holomorphic Function with Polar Singularities”, Indiana Univ. Math. J., 52:6 (2003), 1663–1680  crossref  mathscinet  zmath  isi
    4. S. A. Imomkulov, “On holomorphic continuation of functions defined on a pencil of boundary complex lines”, Izv. Math., 69:2 (2005), 345–363  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. A. Sadullaev, Z. Ibragimov, “The class $R$ and finely analytic functions”, Sb. Math., 209:8 (2018), 1234–1247  mathnet  crossref  crossref  adsnasa  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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