On uniform approximation of harmonic functions

Mazalov, M.

doi:10.1090/S1061-0022-2012-01215-X

On uniform approximation of harmonic functions
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by M. Ya. Mazalov
Translated by: S. Kislyakov

St. Petersburg Math. J. 23 (2012), 731-759

DOI: https://doi.org/10.1090/S1061-0022-2012-01215-X

Published electronically: April 13, 2012

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Abstract:

The paper is devoted to uniform approximation by harmonic functions on compact sets. The result is an approximation theorem for an individual function under the condition that, on the complement to the compact set, the harmonic capacity is “homogeneous” in a sense. The proof involves a refinement of Vitushkin’s localization method.

References

M. V. Keldysh, On the solvability and the stability of the Dirichlet problem, Uspekhi Mat. Nauk vyp. 8 (1941), 171–231. (Russian)
Jacques Deny, Systèmes totaux de fonctions harmoniques, Ann. Inst. Fourier (Grenoble) 1 (1949), 103–113 (1950) (French). MR 37414
A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141–199 (Russian). MR 0229838
A. G. O’Farrell, Uniform approximation by harmonic functions, Problem Book 3. Part 2, Lecture Notes in Math., vol. 1574, Springer-Verlag, 1994, Problem 12.15, p. 121.
P. V. Paramonov, Harmonic approximations in the $C^1$-norm, Mat. Sb. 181 (1990), no. 10, 1341–1365 (Russian); English transl., Math. USSR-Sb. 71 (1992), no. 1, 183–207. MR 1085885, DOI 10.1070/SM1992v071n01ABEH002129
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Reese Harvey and John C. Polking, A notion of capacity which characterizes removable singularities, Trans. Amer. Math. Soc. 169 (1972), 183–195. MR 306740, DOI 10.1090/S0002-9947-1972-0306740-4
M. Ya. Mazalov, A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations, Mat. Sb. 199 (2008), no. 1, 15–46 (Russian, with Russian summary); English transl., Sb. Math. 199 (2008), no. 1-2, 13–44. MR 2410145, DOI 10.1070/SM2008v199n01ABEH003909
M. Ya. Mazalov, On uniform approximations by bi-analytic functions on arbitrary compact sets in $\Bbb C$, Mat. Sb. 195 (2004), no. 5, 79–102 (Russian, with Russian summary); English transl., Sb. Math. 195 (2004), no. 5-6, 687–709. MR 2091640, DOI 10.1070/SM2004v195n05ABEH000822
Joan Verdera, On the uniform approximation problem for the square of the Cauchy-Riemann operator, Pacific J. Math. 159 (1993), no. 2, 379–396. MR 1214077
P. V. Paramonov, Some new criteria for the uniform approximability of functions by rational fractions, Mat. Sb. 186 (1995), no. 9, 97–112 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 9, 1325–1340. MR 1360189, DOI 10.1070/SM1995v186n09ABEH000070
Dzh. Verdera, M. S. Mel′nikov, and P. V. Paramonov, $C^1$-approximation and the extension of subharmonic functions, Mat. Sb. 192 (2001), no. 4, 37–58 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 3-4, 515–535. MR 1834090, DOI 10.1070/sm2001v192n04ABEH000556
Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986
Reese Harvey and John Polking, Removable singularities of solutions of linear partial differential equations, Acta Math. 125 (1970), 39–56. MR 279461, DOI 10.1007/BF02838327
N. N. Tarkhanov, Ryad Lorana dlya resheniĭ èllipticheskikh sistem, “Nauka” Sibirsk. Otdel., Novosibirsk, 1991 (Russian, with Russian summary). MR 1226897
P. V. Paramonov and K. Yu. Fedorovskiĭ, On uniform and $C^1$-approximability of functions on compact sets in $\textbf {R}^2$ by solutions of second-order elliptic equations, Mat. Sb. 190 (1999), no. 2, 123–144 (Russian, with Russian summary); English transl., Sb. Math. 190 (1999), no. 1-2, 285–307. MR 1701003, DOI 10.1070/SM1999v190n02ABEH000386
N. S. Landkof, Osnovy sovremennoĭ teorii potentsiala, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0214795
Guy David, Wavelets and singular integrals on curves and surfaces, Lecture Notes in Mathematics, vol. 1465, Springer-Verlag, Berlin, 1991. MR 1123480, DOI 10.1007/BFb0091544
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
Joan Mateu and Joan Orobitg, Lipschitz approximation by harmonic functions and some applications to spectral synthesis, Indiana Univ. Math. J. 39 (1990), no. 3, 703–736. MR 1078735, DOI 10.1512/iumj.1990.39.39035
V. S. Vladimirov, Uravneniya matematicheskoĭ fiziki, 5th ed., “Nauka”, Moscow, 1988 (Russian). MR 978200
Joan Verdera, Removability, capacity and approximation, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 419–473. MR 1332967