Let $b\in \mathbb{R}\neq 0$, and consider the *translation operator*:
$$
\begin{align}
T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\
 f &\mapsto f(\cdot + b).
\end{align}
$$
*Are there known characterization of $T_b$-[*cyclic*][1] functions $f$; i.e.: $\overline{\text{span}(\{T_b^n(f)\}_{n=0}^{\infty})}=C(\mathbb{R})$?*


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**What I'm looking for:**  When $C(\mathbb{R})$ is replaced by $L^1(\mathbb{R})$ then, the [Wiener's Tauberian theorem][2] gives a characterization of functions $f\in L^1(\mathbb{R})$ as precisely those $f\in L^1(\mathbb{R})$ for which $\hat{f}(x)\neq 0$ for all $x \in \mathbb{R}$ (where $\hat{f}$ denote the Fourier-transform of $f$).  Is there an analogous "simple" criterion/characterization in the case of translation operators on $C(\mathbb{R})$?


  [1]: https://encyclopediaofmath.org/wiki/Cyclic_vector
  [2]: https://en.wikipedia.org/wiki/Wiener's_Tauberian_theorem