Carleman approximation on Riemann surfaces. (English) Zbl 0583.30035
A closed subset E of a non-compact (connected) Riemann surface R is called a set of holomorphic (respectively meromorphic) Carleman approximation if whenever f is continuous on E and holomorphic on the interior \(E^ 0\) of E and \(\epsilon\) is continuous and positive on E, there exists a holomorphic (respectively meromorphic) function g on R such that
\[
| f(p)-g(p)| <\epsilon (p),\quad for\quad all\quad p\in E.
\]
A generalization of a result of A. H. Nersesyan [Izv. Akad. Nauk Arm. SSR, Ser. Mat. 6, 465-471 (1971; Zbl 0235.30041)] obtained previously in the case of the complex plane enable us to give a complete topological characterization of the sets of holomorphic Carleman approximation and to give a sufficient condition on the sets of meromorphic Carleman approximation in terms of Gleason parts. It is also shown that a necessary condition on the components of the interior of the sets of Carleman approximation introduced by P. M. Gauthier [Izv. Akad. Nauk Arm. SSR, Ser. Mat. 4, 319-326 (1969; Zbl 0189.360)] must also hold for the components of the fine interior.
MSC:
| 30E10 | Approximation in the complex plane |
References:
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