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

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

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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• http://mi.mathnet.ru/eng/msb386
• https://doi.org/10.4213/sm386
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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. K. Yu. Fedorovskiy, “Approximation and Boundary Properties of Polyanalytic Functions”, Proc. Steklov Inst. Math., 235 (2001), 251–260
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
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
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
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
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
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
8. Konstantin Yu. Fedorovskiy, “C m -Approximation by Polyanalytic Polynomials on Compact Subsets of the Complex Plane”, Complex anal oper theory, 2010
9. M. Ya. Mazalov, “Uniform approximation problem for harmonic functions”, St. Petersburg Math. J., 23:4 (2012), 731–759
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
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
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
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
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
15. A. B. Zaitsev, “Mappings by the Solutions of Second-Order Elliptic Equations”, Math. Notes, 95:5 (2014), 642–655
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
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
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
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
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
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