# Density of $\mathrm{span} \{ f(\cdot - n) : n \in \Bbb Z \}$ in $C(\mathbb R)$

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!

• 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). Jan 19 at 22:17
• en.wikipedia.org/wiki/Wiener%27s_Tauberian_theorem Jan 20 at 6:30
• The translation operator is thus cyclic. There is a lot of literature about cyclic, supercyclic or even hypercyclic opearotors. Jan 20 at 6:50