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I recently came across a survey paper by Allan Pinkus. This paper contains the following result:

Suppose that $g \in L^1(\mathbb R)$ has support in an interval of length at most $2\pi$. If $f=\hat g$ is the Fourier transform of $g$ then $$ \mathrm{span} \{ f(\cdot - n) : n \in \mathbb{Z} \} $$ is dense in $C(\mathbb R)$ (in the topology of uniform convergence on compact sets).

I was wondering if similar results holds for other types of functions. Is there a theory about such functions or is someone of you aware of papers dicussing results in this direction?

After the above result Pinkus mentions that this is a "simple example within a general theory" and refers to papers by Atzmon and Nikolski. However, as I see it, these papers discuss the existence of functions with the above property (in more general spaces than $C(\mathbb R)$) rather than explicit examples of functions with the above property.

Thanks in advance for any help!

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    $\begingroup$ I think it's fairly easy to see that, for $f \in L^2(\mathbb{R})$, the vector space generated by the translates of $f$ (by all real numbers) is dense in $L^2$ iff the Fourier transform $\hat f$ (in the $L^2$ sense) is nonzero a.e. I can write down details if you want (but there's nothing smart or surprising). $\endgroup$
    – Gro-Tsen
    Jan 19 at 22:17
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    $\begingroup$ en.wikipedia.org/wiki/Wiener%27s_Tauberian_theorem $\endgroup$
    – Nick S
    Jan 20 at 6:30
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    $\begingroup$ The translation operator is thus cyclic. There is a lot of literature about cyclic, supercyclic or even hypercyclic opearotors. $\endgroup$ Jan 20 at 6:50

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