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Uspekhi Mat. Nauk, 2012, Volume 67, Issue 6(408), Pages 53–100 (Mi umn9498)  

This article is cited in 22 scientific papers (total in 22 papers)

Conditions for $C^m$-approximability of functions by solutions of elliptic equations

M. Ya. Mazalova, P. V. Paramonovb, K. Yu. Fedorovskiyc

a Smolensk Branch of the Moscow Power Engineering Institute
b Moscow State University
c Bauman Moscow State Technical University

Abstract: This paper is a survey of results obtained over the past 20–30 years in the qualitative theory of approximation of functions by holomorphic, harmonic, and polyanalytic functions (and, in particular, by corresponding polynomials) in the norms of Whitney-type spaces $C^m$ on compact subsets of Euclidean spaces.
Bibliography: 120 titles.

Keywords: $C^m$-approximation by holomorphic, harmonic, and polyanalytic functions; $C^m$-analytic and $C^m$-harmonic capacity; $s$-dimensional Hausdorff content; Vitushkin localization operator; Nevanlinna domains; Dirichlet problem.


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English version:
Russian Mathematical Surveys, 2012, 67:6, 1023–1068

Bibliographic databases:

UDC: 517.53
MSC: Primary 30E10; Secondary 31A05, 31A30, 31A35, 30C20
Received: 18.10.2012

Citation: M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskiy, “Conditions for $C^m$-approximability of functions by solutions of elliptic equations”, Uspekhi Mat. Nauk, 67:6(408) (2012), 53–100; Russian Math. Surveys, 67:6 (2012), 1023–1068

Citation in format AMSBIB
\by M.~Ya.~Mazalov, P.~V.~Paramonov, K.~Yu.~Fedorovskiy
\paper Conditions for $C^m$-approximability of functions by solutions of elliptic equations
\jour Uspekhi Mat. Nauk
\yr 2012
\vol 67
\issue 6(408)
\pages 53--100
\jour Russian Math. Surveys
\yr 2012
\vol 67
\issue 6
\pages 1023--1068

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    This publication is cited in the following articles:
    1. A. L. Volberg, V. Ya. Èiderman, “Non-homogeneous harmonic analysis: 16 years of development”, Russian Math. Surveys, 68:6 (2013), 973–1026  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. 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
    3. M. Ya. Mazalov, “An example of a non-rectifiable Nevanlinna contour”, St. Petersburg Math. J., 27:4 (2016), 625–630  mathnet  crossref  mathscinet  isi  elib
    4. A. D. Baranov, J. J. Carmona, K. Yu. Fedorovskiy, “Density of certain polynomial modules”, Journal of Approximation Theory, 2016  crossref  mathscinet  scopus
    5. K. Yu. Fedorovskiy, “On the density of certain modules of polyanalytic type in spaces of integrable functions on the boundaries of simply connected domains”, Sb. Math., 207:1 (2016), 140–154  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. E. V. Borovik, K. Yu. Fedorovskiy, “On the Relationship Between Nevanlinna and Quadrature Domains”, Math. Notes, 99:3 (2016), 460–464  mathnet  crossref  crossref  mathscinet  isi  elib
    7. V. I. Danchenko, “Cauchy and Poisson formulas for polyanalytic functions and applications”, Russian Math. (Iz. VUZ), 60:1 (2016), 11–21  mathnet  crossref  isi
    8. P. V. Paramonov, “New Criteria for Uniform Approximability by Harmonic Functions on Compact Sets in $\mathbb R^2$”, Proc. Steklov Inst. Math., 298 (2017), 201–211  mathnet  crossref  crossref  isi  elib
    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. M. Ya. Mazalov, “On Nevanlinna domains with fractal boundaries”, St. Petersburg Math. J., 29:5 (2018), 777–791  mathnet  crossref  mathscinet  isi  elib
    11. A. D. Baranov, K. Yu. Fedorovskiy, “On $L^1$-estimates of derivatives of univalent rational functions”, J. Anal. Math., 132 (2017), 63–80  crossref  mathscinet  zmath  isi
    12. Yu. S. Belov, K. Yu. Fedorovskiy, “Model spaces containing univalent functions”, Russian Math. Surveys, 73:1 (2018), 172–174  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. 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
    14. M. Ya. Mazalov, “On Bianalytic Capacities”, Math. Notes, 103:4 (2018), 672–677  mathnet  crossref  crossref  isi  elib
    15. P. V. Paramonov, “Criteria for the individual $C^m$-approximability of functions on compact subsets of $\mathbb R^N$ by solutions of second-order homogeneous elliptic equations”, Sb. Math., 209:6 (2018), 857–870  mathnet  crossref  crossref  adsnasa  isi  elib
    16. K. Yu. Fedorovskiy, “Carathéodory sets and analytic balayage of measures”, Sb. Math., 209:9 (2018), 1376–1389  mathnet  crossref  crossref  adsnasa  isi  elib
    17. 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
    18. Belov Yu. Borichev A. Fedorovskiy K., “Nevanlinna Domains With Large Boundaries”, J. Funct. Anal., 277:8 (2019), 2617–2643  crossref  isi
    19. Mazalov M.Ya., “Bianalytic Capacities and Calderon Commutators”, Anal. Math. Phys., 9:3 (2019), 1099–1113  crossref  isi
    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  isi
    21. P. V. Paramonov, K. Yu. Fedorovskiy, “On $C^m$-reflection of harmonic functions and $C^m$-approximation by harmonic polynomials”, Sb. Math., 211:8 (2020), 1159–1170  mathnet  crossref  crossref
    22. M. Ya. Mazalov, “A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients”, Sb. Math., 211:9 (2020), 1267–1309  mathnet  crossref  crossref
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