Dense integer translates of a real-valued function with unequal limits at infinity

This is a follow up on a Previous question. Let $W$ be the space of continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$\lim_{x\rightarrow \infty} f(x)=0~\mbox{and}~\lim_{x\rightarrow -\infty} f(x) \in \mathbb{R},$$ and consider the sup-norm topology on $W$.

Problem. does there exist $f\in W$ such that the integer translations of $f$ (i.e., 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: $$\sup_{x\in \mathbb{R}} \left |g(x)-\sum_{i \in \Lambda} c_if(x+i) \right |<\epsilon.$$

• Did you read the paper of Atzmon and Olevski which solved your previous problem, and think about whether their techniques would work here? – Nik Weaver Feb 20 '18 at 17:27
• Yes, I read their article. I didn't see a way to use their techniques directly. If such a $g \in W$ for this problem exists, then $f(x)=g(x+1)-g(x)$ has dense integer translates in their setting. In particular $f$ can't be $L^1$. I tried to arrive at a contradiction using this fact. – Marco Feb 20 '18 at 18:01