Geodesic
Criterion of uniform approximability by harmonic functions on compact sets in $\mathbb R^3$
Informatics and Automation, Analytic and geometric issues of complex analysis, Tome 279 (2012), pp. 120-165.

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For a function continuous on a compact set $X\subset\mathbb R^3$ and harmonic inside $X$, we obtain a criterion of uniform approximability by functions harmonic in a neighborhood of $X$ in terms of the classical harmonic capacity. The proof is based on an improved localization scheme of A. G. Vitushkin, on a special geometric construction, and on the methods of the theory of singular integrals. 
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M. Ya. Mazalov. Criterion of uniform approximability by harmonic functions on compact sets in $\mathbb R^3$. Informatics and Automation, Analytic and geometric issues of complex analysis, Tome 279 (2012), pp. 120-165. http://geodesic.mathdoc.fr/item/TRSPY_2012_279_a9/

[1] Khausdorf F., Teoriya mnozhestv, ONTI, M.–L., 1937

[2] Verdera Dzh., Melnikov M. S., Paramonov P. V., “$C^1$-approksimatsiya i prodolzhenie subgarmonicheskikh funktsii”, Mat. sb., 192:4 (2001), 37–58 | DOI | MR | Zbl

[3] Paramonov P. V., “Nekotorye novye kriterii ravnomernoi priblizhaemosti funktsii ratsionalnymi drobyami”, Mat. sb., 186:9 (1995), 97–112 | MR | Zbl

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

[5] O'Farrell A. G., “Uniform approximation by harmonic functions”, Linear and complex analysis, Problem book 3. Part II, Lect. Notes Math., 1574, Springer, Berlin, 1994, 121

[6] Mazalov M. Ya., “O zadache ravnomernogo priblizheniya garmonicheskikh funktsii”, Algebra i analiz, 23:4 (2011), 136–178 | MR

[7] Mazalov M. Ya., “O ravnomernom priblizhenii garmonicheskimi funktsiyami na kompaktakh v $\mathbb R^3$”, Zap. nauch. sem. POMI, 389, 2011, 162–190 | MR

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

[9] Landkof N. S., Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR | Zbl

[10] Keldysh M. V., “O razreshimosti i ustoichivosti zadachi Dirikhle”, DAN, 18:6 (1938), 315–318 | Zbl

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

[12] Labrèche M., De l'approximation harmonique uniforme, Thèse, Univ. Montréal, 1982

[13] Gonchar A. A., “O ravnomernom priblizhenii nepreryvnykh funktsii garmonicheskimi”, Izv. AN SSSR. Ser. mat., 27:6 (1963), 1239–1250 | MR | Zbl

[14] Lysenko Yu. A., Pisarevskii B. M., “Neustoichivost garmonicheskoi emkosti i priblizheniya nepreryvnykh funktsii garmonicheskimi”, Mat. sb., 76(118):1 (1968), 52–71 | MR | Zbl

[15] Debiard A., Gaveau B., “Potentiel fin et algèbres de fonctions analytiques. I”, J. Funct. Anal., 16 (1974), 289–304 | DOI | MR | Zbl

[16] Gauthier P. M., Ladouceur S., “Uniform approximation and fine potential theory”, J. Approx. Theory, 72:2 (1993), 138–140 | DOI | MR | Zbl

[17] Fuglede B., Finely harmonic functions, Lect. Notes Math., 289, Springer, Berlin, 1972 | MR | Zbl

[18] Mazalov M. Ya., “Kriterii ravnomernoi priblizhaemosti na proizvolnykh kompaktakh dlya reshenii ellipticheskikh uravnenii”, Mat. sb., 199:1 (2008), 15–46 | DOI | MR | Zbl

[19] Kolesnikov S. V., “Ob odnoi teoreme M. V. Keldysha, kasayuscheisya potochechnoi skhodimosti posledovatelnostei polinomov”, Mat. sb., 124(166):4 (1984), 568–570 | MR | Zbl

[20] Mazalov M. Ya., “O ravnomernykh priblizheniyakh bianaliticheskimi funktsiyami na proizvolnykh kompaktakh v $\mathbb C$”, Mat. sb., 195:5 (2004), 79–102 | DOI | MR | Zbl

[21] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1988 | MR

[22] Harvey R., Polking J., “A notion of capacity which characterizes removable singularities”, Trans. Amer. Math. Soc., 169 (1972), 183–195 | DOI | MR | Zbl

[23] Karleson L., Izbrannye problemy teorii isklyuchitelnykh mnozhestv, Mir, M., 1971 | MR | Zbl

[24] O'Farrell A. G., “Metaharmonic approximation in Lipschitz norms”, Proc. R. Ir. Acad. A, 75 (1975), 317–330 | MR | Zbl

[25] Harvey R., Polking J., “Removable singularities of solutions of linear partial differential equations”, Acta math., 125 (1970), 39–56 | DOI | MR | Zbl

[26] Tarkhanov N. N., Ryad Lorana dlya reshenii ellipticheskikh sistem, Nauka, Novosibirsk, 1991 | MR | Zbl

[27] Paramonov P. V., “O garmonicheskikh approksimatsiyakh v $C^1$-norme”, Mat. sb., 181:10 (1990), 1341–1365 | MR | Zbl

[28] Stein I. M., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR | Zbl

[29] Verdera J., “Removability, capacity and approximation”, Complex potential theory, Proc. NATO Adv. Study Inst. Sémin. math. super. (Montréal, 1993), NATO Adv. Sci. Int. Ser. C: Math. Phys. Sci., 439, Kluwer, Dordrecht, 1994, 419–473 | MR | Zbl

[30] Mateu J., Orobitg J., “Lipshitz approximation by harmonic functions and some applications to spectral synthesis”, Indiana Univ. Math. J., 39 (1990), 703–736 | DOI | MR | Zbl

[31] David G., Wavelets and singular integrals on curves and surfaces, Lect. Notes Math., 1465, Springer, Berlin, 1991 | DOI | MR