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A Runge theorem for harmonic functions on closed subsets of Riemann surfaces. (English) Zbl 0656.31001

Let F be a closed subset of an open Riemann surface S. It is shown that every harmonic function in a neighborhood of F can be uniformly approximated by essentially harmonic functions on S: that is, harmonic functions with “poles” of logarithmic type. This result generalizes work of P. M. Gauthier, M. Goldstein and W. H. Ow [Trans. Am. Math. Soc. 261, 169-183 (1980; Zbl 0447.30035)] on Riemann surfaces of finite genus.
Reviewer: S.Ladouceur

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30F15 Harmonic functions on Riemann surfaces
30E10 Approximation in the complex plane

Citations:

Zbl 0447.30035
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References:

[1] A. Boivin and P. M. Gauthier, Approximation harmonique sur les surfaces de Riemann, Canad. J. Math. 36 (1984), no. 1, 1 – 8 (French). · Zbl 0509.31003
[2] A. Dufresnoy, P. M. Gauthier, and W. H. Ow, Uniform approximation on closed sets by solutions of elliptic partial differential equations, Complex Variables Theory Appl. 6 (1986), no. 2-4, 235 – 247. · Zbl 0569.35022
[3] P. M. Gauthier, M. Goldstein, and W. H. Ow, Uniform approximation on unbounded sets by harmonic functions with logarithmic singularities, Trans. Amer. Math. Soc. 261 (1980), no. 1, 169 – 183. · Zbl 0447.30035
[4] P. M. Gauthier and W. Hengartner, Uniform approximation on closed sets by functions analytic on a Riemann surface, Approximation theory (Proc. Conf. Inst. Math., Adam Mickiewicz Univ., Poznań, 1972) Reidel, Dordrecht, 1975, pp. 63 – 69. · Zbl 0322.30034
[5] Paul M. Gauthier and Walter Hengartner, Approximation uniforme qualitative sur des ensembles non bornés, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 82, Presses de l’Université de Montréal, Montreal, Que., 1982 (French). · Zbl 0482.30028
[6] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. · Zbl 0199.40603
[7] R. Narasimhan, Analysis on real and complex manifolds, North-Holland Mathematical Library, vol. 35, North-Holland Publishing Co., Amsterdam, 1985. Reprint of the 1973 edition. · Zbl 0583.58001
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