Universal Entire Functions. Prove that there exists an entire function with the following "universal" property:
Given any entire function $h$, there is an increasing sequence $\{ N_{k}\}_{k=1}^{\infty}$ of positive integers such that $$\lim_{k \rightarrow \infty} F(z + N_{k}) = h(z)$$
Is Runge approximation theorem implicated ?
This is a famous theorem by G. Birkhoff (1929):
G. Birkhoff, Démonstration d'un théorème elémentaire sur les fonctions entiéres, Cr. R. Acad. Sci. Paris, 189 (1929), 473-475.
Sketch of proof. Let $\{p_j(z)\}_{j\in\mathbb N}$ be a dense (locally uniformly) sequence of polynomials in $\mathcal H(\mathbb C)$ and assume that each $p_j$ occurs infinitely often in this sequence. Let $\{D_j\}_{j\in\mathbb N}$ be a sequence of disjoint closed discs, each $D_j$ of radius $j$, such that the centers $\{c_j\}_{n\in\mathbb N}$ form an increasing sequence on the positive real axis. Let $\{E_j\}_{j\in\mathbb N}$ be a sequence of closed discs, each centered at the origin, such that $D_j \subset E_j$ and $D_{j+1} \cap E_j = \varnothing$.
Thus, $D_i\subset E_j$ for $1 \le i \le j$ and $D_i \cap E_j =\varnothing$, for $i \ge j + 1$.
Call $q_1 = q_1$. By Runge's theorem, there is a polynomial $q_2$, such that $\|q_2\|_{E_1} < 1/2$ (here $\|f\|_K=\sup_{z\in K}|f(z)|$) and $$ \big|q_2(z) - \big( p_2(z-c_2) - q_1(z)\big)\big| <1/2, $$ on $D_2$. Next, choose a polynomial $q_3$, such that $\|q_3\|_{E_2} <\frac{1}{4}$, and $$ \big|q_3(z) - \big(p_3(z- c_3) - q_1(z) - q_2(z)\big)\big| <\frac{1}{4}, $$ on $D_3$ In general, let $q_n$ be a polynomial, such that $\|q_n\|_{E_{n-1}} < \frac{1}{2^{n-1}}$ and $$ \bigg|\,q_n(z) - \bigg(p_n(z - c_n) - \sum_{i=1}^{n-1}q_i(z)\bigg)\bigg|<\frac{1}{2^{n-1}}, $$ on $D_n$.
We shall show that the $f = \sum_{n=1}^\infty q_n$ does what we want. It is readily seen that $f$ is entire. It remains to show that if $g \in\mathcal H(\mathbb C)$, and for $R > 0$ and $\varepsilon > 0$, arbitrary, then for some $j$; $$ |f(z + c_j) - g(z)| < 2, $$ whenever $|z|< R$ In fact, it suffices to show this for $g=p$, where $p=p_k$, for some $k$. Noting that there are infinitely many $k$, for which $p = p_k$, we can choose such a large $k$ so that $$ \bigg\|\,f - \sum_{i=1}^k q_i\,\bigg\|_{E_{k-1}}. $$ Also, $$ \bigg|\sum_{i=1}^k q_i(z) - p(z - c_k)\bigg| < \varepsilon, $$ for all $z \in D_k$. In other words, for all $z \in D_k$, we have that $$ |f(z) - p(z - c_k)| < 2\varepsilon, $$ and the result follows by a change of variable.