The Wiener Tauberian Theorem gives condition on an $f\in L^1(\mathbb{R})$ such that the "induced 1-parameter family" $\{T_b(f)\}_{b\in \mathbb{R}}$ has a dense span in $L^1(\mathbb{R})$; where $T_b$ is the Group of transation operators on $L^1(\mathbb{R})$ defined by $f\mapsto T_b(\mathbb{R})$.
Are there other "similar results" stating that: Given a group (under composition) $(T_t)_{t\in \mathbb{R}}$ of (continuous linear) operators on $L^1(\mathbb{R})$ then: $$ \mbox{If $f\in L^1(\mathbb{R})$ satisfies ___ then $\overline{\operatorname{span}(\{T_t(f)\}_{t\in\mathbb{R}}}=L^1(\mathbb{R})$.} $$
Note: I would be interested (though less so) in results where either:
- $L^1(\mathbb{R})$ is replaced by $C(\mathbb{R})$ with the compact-convergence topology.
- $(T_t)_{t}$ forms a semi-group,
- If $\mathbb{R}$ is replaced by a locally-compact topological group $G$ and the Lebesgue measure by the Haar measure on $G$
Related Post: Here.
(What I'm not looking for): I just note that conditions on $f$ which I would not be interested in, are those relating to hyper/super-cyclicity. This is because, such conditions need $f$ to be "infinitely complicated" (see this paper) while I'm looking for "tractable/simple $f$" (as in Wiener's case).
