on uniform approximation by entire functions
In order that there exist for any continuous complex-valued function $f(z)$ on a continuum $E$ and any rapidly-decreasing positive function $\epsilon(r)$, $0\leq r$ (as $r\to\infty$), having a positive lower bound on any finite interval, an entire function $g(z)$ such that
$$|f(z)-g(z)|<\epsilon(|z|),\quad z\in E,$$
it is necessary and sufficient that $E$ has no interior points and that there exists a function $\eta(t)$, $0<t<+\infty$, that increases to $+\infty$ and is such that any point $z$ of the complement $CE$ can be joined to $\infty$ by a Jordan curve situated outside $E$ and outside the disc $|\zeta|<\eta(|z|)$.
|||M.V. Keldysh, M.A. Lavrent'ev, "Sur un problème de M. Carleman" Dokl. Akad. Nauk SSSR , 23 : 8 (1939) pp. 746–748|
|||S.N. Mergelyan, "Uniform approximation to functions of a complex variable" Transl. Amer. Math. Soc. , 3 (1962) pp. 294–391 Uspekhi Mat. Nauk , 7 : 2 (1952) pp. 31–1A2|
|[a1]||D. Gaier, "Lectures on complex approximation" , Birkhäuser (1987) (Translated from German)|
Keldysh–Lavrent'ev theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Keldysh%E2%80%93Lavrent%27ev_theorem&oldid=22644