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Mat. Sb., 2008, Volume 199, Number 1, Pages 15–46 (Mi msb3884)  

This article is cited in 12 scientific papers (total in 12 papers)

A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations

M. Ya. Mazalov

Military Academy of Air Defence Forces of Russia Federation named after A. M. Vasilevskii

Abstract: Let $X$ be an arbitrary compact subset of the plane. It is proved that if $L$ is a homogeneous elliptic operator with constant coefficients and locally bounded fundamental solution, then each function $f$ that is continuous on $X$ and satisfies the equation $Lf=0$ at all interior points of $X$ can be uniformly approximated on $X$ by solutions of the same equation having singularities outside $X$. A theorem on uniform piecemeal approximation of a function is also established under weaker constraints than in the standard Vitushkin scheme.
Bibliography: 24 titles.

DOI: https://doi.org/10.4213/sm3884

Full text: PDF file (727 kB)
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English version:
Sbornik: Mathematics, 2008, 199:1, 13–44

Bibliographic databases:

UDC: 517.538.5+517.956.2
MSC: Primary 41A30; Secondary 30E10, 35J99
Received: 22.05.2007

Citation: M. Ya. Mazalov, “A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations”, Mat. Sb., 199:1 (2008), 15–46; Sb. Math., 199:1 (2008), 13–44

Citation in format AMSBIB
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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. M. Ya. Mazalov, “Uniform approximation by harmonic functions on compact subsets of $\mathbb R^3$”, J. Math. Sci. (N. Y.), 182:5 (2012), 674–689  mathnet  crossref
    2. M. Ya. Mazalov, “Uniform approximation problem for harmonic functions”, St. Petersburg Math. J., 23:4 (2012), 731–759  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    3. 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
    4. 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
    5. M. Ya. Mazalov, “Criterion of uniform approximability by harmonic functions on compact sets in $\mathbb R^3$”, Proc. Steklov Inst. Math., 279 (2012), 110–154  mathnet  crossref  mathscinet  isi  elib
    6. Fedorovskii K.Yu., “O ravnomernoi approksimatsii funktsii na ploskikh kompaktakh resheniyami odnorodnykh ellipticheskikh uravnenii”, Vest. Mosk. gos. tekhnich. un-ta im. N. E. Baumana. Ser. Estestvennye nauki, 2012, no. 3, 3–15  elib
    7. M. Ya. Mazalov, “O ravnomernoi priblizhaemosti resheniyami ellipticheskikh uravnenii poryadka vyshe dvukh”, Ufimsk. matem. zhurn., 4:4 (2012), 108–118  mathnet
    8. A.D. Baranov, J.J. Carmona, K.Yu. Fedorovskiy, “Density of certain polynomial modules”, Journal of Approximation Theory, 2015  crossref  mathscinet  scopus
    9. 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
    10. Fedorovskiy K.Yu., “Two Problems on Approximation By Solutions of Elliptic Systems on Compact Sets in the Plane”, Complex Var. Elliptic Equ., 63:7-8, SI (2018), 961–975  crossref  mathscinet  zmath  isi  scopus
    11. M. Ya. Mazalov, “On Bianalytic Capacities”, Math. Notes, 103:4 (2018), 672–677  mathnet  crossref  crossref  isi  elib
    12. 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
  • Математический сборник Sbornik: Mathematics (from 1967)
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