Keldysh-Lavrent'ev theorem

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 $|\zeta|<\eta(|z|)$.

This result of M.V. Keldysh and M.A. Lavrent'ev [1] summarizes numerous investigations on approximation by entire functions initiated by the Carleman theorem (Section 3; see also [2]).

References

[1]	M.V. Keldysh, M.A. Lavrent'ev, "Sur un problème de M. Carleman" Dokl. Akad. Nauk SSSR , 23 : 8 (1939) pp. 746–748
[2]	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

References

[a1]	D. Gaier, "Lectures on complex approximation" , Birkhäuser (1987) (Translated from German)