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Mat. Sb., 1999, Volume 190, Number 2, Pages 123–144 (Mi msb386)  

This article is cited in 20 scientific papers (total in 20 papers)

Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations

P. V. Paramonova, K. Yu. Fedorovskiyb

a M. V. Lomonosov Moscow State University
b Institute of Information Systems in Management at the State University of Management

Abstract: Several necessary and sufficient conditions for the existence of uniform or $C^1$-approximation of functions on compact subsets of $\mathbb R^2$ by solutions of elliptic systems of the form $c_{11}u_{x_1x_1}+2c_{12}u_{x_1x_2}+c_{22}u_{x_2x_2}=0$ with constant complex coefficients $c_{11}$, $c_{12}$ and $c_{22}$ are obtained. The proofs are based on a refinement of Vitushkin's localization method, in which one constructs localized approximating functions by “gluing together” some special many-valued solutions of the above equations. The resulting conditions of approximation are of a topological and metric nature.

DOI: https://doi.org/10.4213/sm386

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English version:
Sbornik: Mathematics, 1999, 190:2, 285–307

Bibliographic databases:

UDC: 517.538.5+517.956.22
MSC: 30E10, 35J15
Received: 21.06.1996 and 02.06.1998

Citation: 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”, Mat. Sb., 190:2 (1999), 123–144; Sb. Math., 190:2 (1999), 285–307

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. K. Yu. Fedorovskiy, “Approximation and Boundary Properties of Polyanalytic Functions”, Proc. Steklov Inst. Math., 235 (2001), 251–260  mathnet  mathscinet  zmath
    2. A. B. Zaitsev, “Uniform Approximability of Functions by Polynomials of Special Classes on Compact Sets in $\mathbb R^2$”, Math. Notes, 71:1 (2002), 68–79  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. J. J. Carmona, P. V. Paramonov, K. Yu. Fedorovskiy, “On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions”, Sb. Math., 193:10 (2002), 1469–1492  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. B. Zaitsev, “Uniform Approximation of Functions by Polynomial Solutions to Second-Order Elliptic Equations on Compact Sets in $\mathbb{R}^2$”, Math. Notes, 74:1 (2003), 38–48  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. A. B. Zaitsev, “Uniform approximability of functions by polynomial solutions of second-order elliptic equations on compact plane sets”, Izv. Math., 68:6 (2004), 1143–1156  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. A. B. Zaitsev, “Uniform Approximation by Polynomial Solutions of Second-Order Elliptic Equations, and the Corresponding Dirichlet Problem”, Proc. Steklov Inst. Math., 253 (2006), 57–70  mathnet  crossref  mathscinet  elib
    7. M. Ya. Mazalov, “A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations”, Sb. Math., 199:1 (2008), 13–44  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. Konstantin Yu. Fedorovskiy, “C m -Approximation by Polyanalytic Polynomials on Compact Subsets of the Complex Plane”, Complex anal oper theory, 2010  crossref  mathscinet  isi  scopus  scopus  scopus
    9. 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
    10. 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
    11. Fedorovskii K.Yu., “O ravnomernoi approksimatsii funktsii na ploskikh kompaktakh resheniyami odnorodnykh ellipticheskikh uravnenii”, Vestnik moskovskogo gosudarstvennogo tekhnicheskogo universiteta im. N.E. Baumana. seriya: estestvennye nauki, 2012, no. 3, 3–15  elib
    12. Fedorovskii K.Yu., “Oblasti i kompakty karateodori v teorii priblizhenii analiticheskimi funktsiyami”, Vestnik moskovskogo gosudarstvennogo tekhnicheskogo universiteta im. N.E. Baumana. seriya: estestvennye nauki, 2012, 36–45  elib
    13. 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  crossref  mathscinet  zmath  isi
    14. 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
    15. A. B. Zaitsev, “Mappings by the Solutions of Second-Order Elliptic Equations”, Math. Notes, 95:5 (2014), 642–655  mathnet  crossref  crossref  mathscinet  isi  elib
    16. M. Ya. Mazalov, P. V. Paramonov, “Criteria for $C^m$-approximability by bianalytic functions on planar compact sets”, Sb. Math., 206:2 (2015), 242–281  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    17. 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
    18. 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
    19. 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
    20. Paramonov P.V. Tolsa X., “On C-1-Approximability of Functions By Solutions of Second Order Elliptic Equations on Plane Compact Sets and C-Analytic Capacity”, Anal. Math. Phys., 9:3 (2019), 1133–1161  crossref  mathscinet  zmath  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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