Let $W$ be the space of continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\lim\_{x\rightarrow \pm \infty} f(x)=0$, and consider the sup-norm topology on $W$. 


**Problem.** does there exist $f\in W$ such that the set of translations of $f$ (i.e., the set of functions $f\_i(x)=f(x+i)$, $i\in \mathbb{Z}$) have a dense linear span in $W$? 


Having a dense linear span means that for every $g\in W$ and every $\epsilon>0$, there exists a finite subset $\Lambda \subseteq \mathbb{Z}$ and real numbers $c\_i \in \mathbb{R}, i\in \Lambda$ such that: 
$$\max\_{x\in \mathbb{R}} \left |g(x)-\sum\_{i \in \Lambda} c\_if(x+i) \right |<\epsilon.$$ 



Set of translations of a real function having a dense linear span
Yes. This was proved by [Atzmon and Olevski.](https://www.sciencedirect.com/science/article/pii/S0021904596901069)


