Geodesic
Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations
Sbornik. Mathematics, Tome 190 (1999) no. 2, pp. 285-307 
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. Sbornik. Mathematics, Tome 190 (1999) no. 2, pp. 285-307. http://geodesic.mathdoc.fr/item/SM_1999_190_2_a5/
@article{SM_1999_190_2_a5,
     author = {P. V. Paramonov and K. Yu. Fedorovskiy},
     title = {Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations},
     journal = {Sbornik. Mathematics},
     pages = {285--307},
     year = {1999},
     volume = {190},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1999_190_2_a5/}
}
TY  - JOUR
AU  - P. V. Paramonov
AU  - K. Yu. Fedorovskiy
TI  - Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations
JO  - Sbornik. Mathematics
PY  - 1999
SP  - 285
EP  - 307
VL  - 190
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1999_190_2_a5/
LA  - en
ID  - SM_1999_190_2_a5
ER  - 
%0 Journal Article
%A P. V. Paramonov
%A K. Yu. Fedorovskiy
%T Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations
%J Sbornik. Mathematics
%D 1999
%P 285-307
%V 190
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1999_190_2_a5/
%G en
%F SM_1999_190_2_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Keldysh M. V., “O razreshimosti i ustoichivosti zadachi Dirikhle”, UMN, 1941, no. 8, 171–231 | MR | Zbl

[2] Deny J., “Systèmes totaux de functions harmoniques”, Ann. Inst. Fourier (Grenoble), 1 (1949), 103–113 | MR

[3] Walsh J. L., “The approximation of harmonic functions by polynomials and by harmonic rational functions”, Bull. Amer. Math. Soc., 35 (1929), 499–544 | DOI | MR | Zbl

[4] Gamelin T., Ravnomernye algebry, Mir, M., 1973 | Zbl

[5] Paramonov P. V., “$C^m$-priblizheniya garmonicheskimi polinomami na kompaktnykh mnozhestvakh v ${\mathbb R}^n$”, Matem. sb., 184:2 (1993), 105–128 | MR | Zbl

[6] Wang J. L., “A localization operator for rational modules”, Rocky Mountain J. Math., 19 (1989), 999–1002 | MR | Zbl

[7] Carmona J. J., “Mergelyan approximation theorem for rational modules”, J. Approx. Theory, 44 (1985), 113–126 | DOI | MR | Zbl

[8] Paramonov P. V., “O priblizheniyakh garmonicheskimi polinomami v $C^1$-norme na kompaktakh v ${\mathbb R}^2$”, Izv. RAN. Ser. matem., 57:2 (1993), 113–124 | MR | Zbl

[9] Verdera J., “Removability, capacity and approximation”, Complex potential theory, NATO ASI Series C, 439, Kluwer Acad. Publ., Dordrecht, 1993, 419–473 | MR

[10] Vitushkin A. G., “Analiticheskaya emkost mnozhestv v zadachakh teorii priblizhenii”, UMN, 22:6 (1967), 141–199 | MR

[11] Bitsadze A. G., Kraevye zadachi dlya ellipticheskikh uravnenii vtorogo poryadka, Nauka, M., 1966

[12] Trev Zh., Lektsii po lineinym uravneniyam v chastnykh proizvodnykh s postoyannymi koeffitsientami, Mir, M., 1965 | Zbl

[13] Verdera J., “$C^m$-approximation by solution of elliptic equations and Calderon–Zygmund operators”, Duke Math. J., 55:1 (1987), 157–187 | DOI | MR | Zbl

[14] Paramonov P. V., Verdera J., “Approximation by solutions of elliptic equations on closed subsets of Euclidean space”, Math. Scand., 74 (1994), 249–259 | MR | Zbl

[15] Mergelyan S. N., “Ravnomernye priblizheniya funktsii kompleksnogo peremennogo”, UMN, 7:2 (1952), 31–122 | MR | Zbl

[16] Fedorovskii K. Yu., “O ravnomernykh priblizheniyakh funktsii $n$-analiticheskimi polinomami na spryamlyaemykh konturakh v ${\mathbb C}$”, Matem. zametki, 59:4 (1996), 604–610 | MR | Zbl

[17] Smirnov S. K., Khavin V. P., “Zadachi priblizheniya i prodolzheniya dlya nekotorykh klassov vektornykh polei”, Algebra i analiz, 10:3 (1998), 133–162 | MR | Zbl

[18] Verdera J., “On the uniform approximation problem for the square of the Cauchy–Riemann operator”, Pasific J. Math., 159 (1993), 379–396 | MR | Zbl