Let $b\in \mathbb{R}\neq 0$, and consider the translation operators: $$ \begin{align} T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\ f &\mapsto f(\cdot + b). \end{align} $$ *Are there known characterization of:
- $T_b$-cyclic functions $f$; i.e.: $\overline{\text{span}(\{T_b^n(f)\}_{n=0}^{\infty})}=C(\mathbb{R})$?*
- Functions $f\in C(\mathbb{R})$ such that $\overline{\text{span}(\{T_b(f)\}_{b \in \mathbb{R}})}=C(\mathbb{R})$?
What I'm looking for: When $C(\mathbb{R})$ is replaced by $L^1(\mathbb{R})$ then, the Wiener's Tauberian theorem 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 $C(\mathbb{R})$?
Note: I equip $\mathbb{C}(\mathbb{R})$ with the compact-covergence, aka. uniform convergence on compacts, aka compact-open topology not the uniform one
