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Mat. Zametki, 1997, Volume 62, Issue 3, Pages 372–382 (Mi mz1619)  

This article is cited in 2 scientific papers (total in 2 papers)

Approximation by harmonic functions in the $C^m$-Norm and harmonic $C^m$-capacity of compact sets in $\mathbb R^n$

Yu. A. Gorokhov

M. V. Lomonosov Moscow State University

Abstract: We study the function $\Lambda^m(X)$, $0<m<1$, of compact sets $X$ in $\mathbb R^n$, $n\ge2$, defined as the distance in the space $C^m(X)\equiv\operatorname{lip}^m(X)$ from the function $|x|^2$ to the subspace $H_m(X)$ which is the closure in $C_m(X)$ of the class of functions harmonic in the neighborhood of $X$ (each function in its own neighborhood). We prove the equivalence of the conditions $\Lambda^m(X)=0$ and $C^m(X)=H^m(X)$. We derive an estimate from above that depends only on the geometrical properties of the set $X$ (on its volume).

DOI: https://doi.org/10.4213/mzm1619

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English version:
Mathematical Notes, 1997, 62:3, 314–322

Bibliographic databases:

UDC: 517.5
Received: 01.11.1995

Citation: Yu. A. Gorokhov, “Approximation by harmonic functions in the $C^m$-Norm and harmonic $C^m$-capacity of compact sets in $\mathbb R^n$”, Mat. Zametki, 62:3 (1997), 372–382; Math. Notes, 62:3 (1997), 314–322

Citation in format AMSBIB
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\by Yu.~A.~Gorokhov
\paper Approximation by harmonic functions in the $C^m$-Norm and harmonic $C^m$-capacity of compact sets in $\mathbb R^n$
\jour Mat. Zametki
\yr 1997
\vol 62
\issue 3
\pages 372--382
\mathnet{http://mi.mathnet.ru/mz1619}
\crossref{https://doi.org/10.4213/mzm1619}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1620062}
\zmath{https://zbmath.org/?q=an:0921.41010}
\transl
\jour Math. Notes
\yr 1997
\vol 62
\issue 3
\pages 314--322
\crossref{https://doi.org/10.1007/BF02360872}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000072500900006}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Voroncov, “Joint Approximations of Distributions in Banach Spaces”, Math. Notes, 73:2 (2003), 168–182  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. M. Voroncov, “Estimates of $C^m$-Capacity of Compact Sets in $\mathbb{R}^N$”, Math. Notes, 75:6 (2004), 751–764  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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