Geodesic
Luzin-type Holomorphic Approximation on Closed Subsets of Open Riemann Surfaces
Canadian mathematical bulletin, Tome 60 (2017) no. 2, pp. 300-308

Voir la notice de l'article provenant de la source Cambridge University Press

It is known that if $E$ is a closed subset of an open Riemann surface $R$ and $f$ is a holomorphic function on a neighbourhood of $E$ , then it is “usually” not possible to approximate $f$ uniformly by functions holomorphic on all of $R$ . We show, however, that for every open Riemann surface $R$ and every closed subset $E\subset R$ , there is closed subset $F\subset E$ that approximates $E$ extremely well, such that every function holomorphic on $F$ can be approximated much better than uniformly by functions holomorphic on $R$ .
DOI : 10.4153/CMB-2016-053-8
Mots-clés : 30E15, 30F99, Carleman approximation, tangential approximation, Myrberg surface 
Gauthier, Paul M.; Sharifi, Fatemeh. Luzin-type Holomorphic Approximation on Closed Subsets of Open Riemann Surfaces. Canadian mathematical bulletin, Tome 60 (2017) no. 2, pp. 300-308. doi: 10.4153/CMB-2016-053-8
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[1] [1] Arakelian, N. U., Uniform and tangential approximations by analytic functions. Amer. Math. Soc. Transi. (2) 122(1984), 85–97. Google Scholar

[2] [2] Boivin, A., Carleman approximation on Riemann surfaces. Math. Ann. 275(1986), no. 1, 57–70. Google Scholar | DOI

[3] [3] Carleman, T., Sur un théorème de Weierstrass. Arkiv. Mat. Astron. Fys. 20(1927), 1–5. Google Scholar

[4] [4] Gauthier, P., Tangential approximation by entire functions and functions holomorphic in a disc. Izv. Akad. Nauk Armjan. SSR Ser. Mat. 4(1969), no. 5, 319–326. Google Scholar

[5] [5] Gauthier, P. M. and Hengartner, W., Uniform approximation on closed sets by functions analytic on a Riemann surface. Approximation theory (Proc. Conf. Inst. Math., Adam Mickiewicz Univ., Poznan, 1972), Reidel, Dordrecht, 1975, pp. 63–69. Google Scholar

[6] [6] Gunning, R. C. and Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann. 174(1967), 103–108. Google Scholar | DOI

[7] [7] Nersesjan, A. A., Carleman sets. (Russian) Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6(1971), no. 6, 465–471. Google Scholar

[8] [8] Scheinberg, S., Uniform approximation by functions analytic on a Riemann surface. Ann. of Math. (2) 108(1978), no. 2, 257–298. Google Scholar | DOI

[9] [9] Scheinberg, S., Noninvariance of an approximation property for closed subsets of Riemann surfaces. Trans. Amer. Math. Soc. 262(1980), no. 1, 245–258. Google Scholar | DOI

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