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Izv. RAN. Ser. Mat., 1993, Volume 57, Issue 2, Pages 113–124 (Mi izv880)  

This article is cited in 8 scientific papers (total in 8 papers)

On approximation by harmonic polynomials in the $C^1$-norm on compact sets in $\mathbf R^2$

P. V. Paramonov


Abstract: It is proved that for an arbitrary compact set $X$ in $\mathbf R^2$ the following conditions are equivalent:
1) for every function $f\in C^1(\mathbf R^2)$, harmonic on $X^0$, and for any $\varepsilon>0$ a harmonic polynomial $p$ can be found such that
$$ \|f-p\|_X<\varepsilon,\qquad \|\nabla(f-p)\|_X<\varepsilon; $$
2) the set $\mathbf R^2\setminus X$ is connected

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1994, 42:2, 321–331

Bibliographic databases:

UDC: 517.5
MSC: Primary 41A10, 41A63; Secondary 31A05
Received: 22.10.1992

Citation: P. V. Paramonov, “On approximation by harmonic polynomials in the $C^1$-norm on compact sets in $\mathbf R^2$”, Izv. RAN. Ser. Mat., 57:2 (1993), 113–124; Russian Acad. Sci. Izv. Math., 42:2 (1994), 321–331

Citation in format AMSBIB
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\by P.~V.~Paramonov
\paper On~approximation by harmonic polynomials in the $C^1$-norm on compact sets in~$\mathbf R^2$
\jour Izv. RAN. Ser. Mat.
\yr 1993
\vol 57
\issue 2
\pages 113--124
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\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1994IzMat..42..321P}
\transl
\jour Russian Acad. Sci. Izv. Math.
\yr 1994
\vol 42
\issue 2
\pages 321--331
\crossref{https://doi.org/10.1070/IM1994v042n02ABEH001539}
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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. P. V. Paramonov, K. Yu. Fedorovskiy, “Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations”, Sb. Math., 190:2 (1999), 285–307  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. K. Yu. Fedorovskiy, “Approximation and Boundary Properties of Polyanalytic Functions”, Proc. Steklov Inst. Math., 235 (2001), 251–260  mathnet  mathscinet  zmath
    3. Konstantin Yu. Fedorovskiy, “C m -Approximation by Polyanalytic Polynomials on Compact Subsets of the Complex Plane”, Complex anal oper theory, 2010  crossref
    4. K. Yu. Fedorovskiy, “On $\mathcal C^m$-approximability of functions by polynomial solutions of elliptic equations on compact plane sets”, St. Petersburg Math. J., 24:4 (2013), 677–689  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    5. Fedorovskiy K.Yu., “Uniform and C-M-Approximation by Polyanalytic Polynomials”, Complex Analysis and Potential Theory, CRM Proceedings & Lecture Notes, 55, ed. Boivin A. Mashreghi J., Amer Mathematical Soc, 2012, 323–329  isi
    6. M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskiy, “Conditions for $C^m$-approximability of functions by solutions of elliptic equations”, Russian Math. Surveys, 67:6 (2012), 1023–1068  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. A. O. Bagapsh, K. Yu. Fedorovskiy, “$C^1$ Approximation of Functions by Solutions of Second-Order Elliptic Systems on Compact Sets in $\mathbb R^2$”, Proc. Steklov Inst. Math., 298 (2017), 35–50  mathnet  crossref  crossref  isi  elib
    8. A. O. Bagapsh, K. Yu. Fedorovskiy, “$C^m$ approximation of functions by solutions of second-order elliptic systems on compact sets in the plane”, Proc. Steklov Inst. Math., 301 (2018), 1–10  mathnet  crossref  crossref  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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