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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

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    $\begingroup$ Wiener's Tauberian theorem, as usually formulated, is about the span of $\{ f(x+a): a\in\mathbb R\}$ being dense, not $\{ f(x+nb): n\in\mathbb N\}$. $\endgroup$ Jun 9, 2021 at 16:55
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    $\begingroup$ If $f \in H^1$ and the span of its translates is dense in $L^2$, then it is also dense in $H^1$. This follows from the core theorem, since this span is invariant under the translation semigroup whose generator has domain $H^1$. In particular, the span of the translates of such a function is dense with respect to the sup-norm. $\endgroup$ Jun 10, 2021 at 11:07
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    $\begingroup$ @GiorgioMetafune How does your comment reconcile with the one of ChristianRemling above? $\endgroup$
    – Yemon Choi
    Jun 10, 2021 at 12:13
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    $\begingroup$ The core theorem is Theorem 1.9 pag. 8 of the book by B. Davies "One-parameter semigroups". $\endgroup$ Jun 10, 2021 at 12:24
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    $\begingroup$ @yemonChoi Of course I am only considering $C_0(R)$, that is continuous functions vanishing at infinity. $\endgroup$ Jun 10, 2021 at 12:32

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