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).

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    $\begingroup$ OK, I convert my answer to an "insightful comment" (first time I go that way :). For the combinatorial side of one-parameter deformations, you can have a look here. For the question of existence of a cyclic vector, there is a bunch of examples in § 4.2 for which you can test the density of the orbits. Do not hesitate to interact (I can provide missing proofs and details on request, of course). $\endgroup$ Jun 12, 2021 at 14:53
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    $\begingroup$ Thanks Gérard, I really do appreciate the comment and I currently looking through your paper. I do feel abit silly, but I don't quite see where density in $L^1(\mathbb{R})$ comes in. (P.s.: Let me just briefly mention that I didn't downvote your earlier answer...so no hard feelings :) ) $\endgroup$
    – ABIM
    Jun 12, 2021 at 14:57
  • $\begingroup$ OK, I understand. The paper does not offer already elaborated statement about density, but, rather, there are a lot of situations and, I think, a setting to explore some type of one-parameter groups (acting on holomorphic or continuous functions). So, ok, do not loose time if it does not fit directly your research thread. $\endgroup$ Jun 12, 2021 at 15:12
  • $\begingroup$ @DuchampGérardH.E. Still thanks a lot, I'll continue reading it and I'll let you know if I have any questions. Thanks again :) $\endgroup$
    – ABIM
    Jun 12, 2021 at 15:14
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    $\begingroup$ In the direction of replacing $\mathbf{R}$ by other locally compact groups, I believe that Wiener's Tauberian theorem works perfectly well for locally compact abelian groups. For non abelian groups, I do not know what is the current state of knowledge. It was apparently open in 1976 whether this holds for groups with polynomial growth, see zbmath.org/?q=an%3A0338.43005. See also Yulia Kuznetsova's answer here for other examples and references: mathoverflow.net/q/84943 $\endgroup$ Jun 14, 2021 at 10:03


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