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Mat. Sb. (N.S.), 1975, Volume 97(139), Number 2(6), Pages 278–300 (Mi msb3652)  

On the approximation of functions of several complex variables on fat compact subsets of $\mathbf C^n$ by polynomials

V. N. Senichkin


Abstract: For a compact set $J\subset\mathbf C^n$, we denote by $P(J)$ the algebra of all functions on $J$ which can be approximated uniformly (on $J$) by polynomials in $n$ complex variables, and by $A(J)$ the algebra of all continuous functions on $J$ which are analytic at the interior points of $J$. We shall say that $J$ is fat if it is the closure of an open set.
In this paper, we consider the problem of approximating functions of several complex variables on fat compact sets with connected interior by polynomials. We prove the following theorems.
Theorem 1. There exists a fat polynomially convex $($holomorphically$)$ contractible compact subset $J$ of $\mathbf C^2$ whose interior is homeomorphic to the four-dimensional open ball and such that $P(J)\ne A(J)$.
Theorem 2. There exists a fat polynomially convex contractible compact subset $J$ of $\mathbf C^3$ whose interior is homeomorphic to the six-dimensional open ball and such that $P(J)\ne A(J)$, although the minimal boundaries of the algebras $P(J)$ and $A(J)$ coincide.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 26:2, 260–279

Bibliographic databases:

UDC: 517.5
MSC: Primary 32E20, 32E30, 46J15; Secondary 32E25, 46J20
Received: 24.01.1975

Citation: V. N. Senichkin, “On the approximation of functions of several complex variables on fat compact subsets of $\mathbf C^n$ by polynomials”, Mat. Sb. (N.S.), 97(139):2(6) (1975), 278–300; Math. USSR-Sb., 26:2 (1975), 260–279

Citation in format AMSBIB
\Bibitem{Sen75}
\by V.~N.~Senichkin
\paper On~the approximation of~functions of several complex variables on fat compact subsets of~$\mathbf C^n$ by polynomials
\jour Mat. Sb. (N.S.)
\yr 1975
\vol 97(139)
\issue 2(6)
\pages 278--300
\mathnet{http://mi.mathnet.ru/msb3652}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=397017}
\zmath{https://zbmath.org/?q=an:0327.32005}
\transl
\jour Math. USSR-Sb.
\yr 1975
\vol 26
\issue 2
\pages 260--279
\crossref{https://doi.org/10.1070/SM1975v026n02ABEH002480}


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