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

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

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

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
2. Bloom T., “On the Convergence in Capacity of Rational Approximants”, Constr. Approx., 17:1 (2001), 91–102
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
4. S. A. Imomkulov, “On holomorphic continuation of functions defined on a pencil of boundary complex lines”, Izv. Math., 69:2 (2005), 345–363
5. A. Sadullaev, Z. Ibragimov, “The class $R$ and finely analytic functions”, Sb. Math., 209:8 (2018), 1234–1247
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