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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: $$\max_{x\in \mathbb{R}} \left |g(x)-\sum_{i \in \Lambda} c_if(x+i) \right |<\epsilon.$$

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Yes. This was proved by Atzmon and Olevski.

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  • $\begingroup$ Thank you! I actually have a follow up question. Does the same result hold for the space of functions from R into R^2? Section 5 of the article you shared mentions a multivariate version in passing, but I am not sure this is what the authors meant. $\endgroup$ – Marco Jan 15 '18 at 18:30

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