TITLE: What functions can be obtained as a convolution of a Schwartz function and a tempered distribution?
QUESTION [6 upvotes]: Let $\mathcal S (\mathbb R)$ denote the space of Schwartz functions on $\mathbb R$ and $\mathcal S^* (\mathbb R)$ denote the dual space of Schwartz (a.k.a tempered)  distributions.
We consider $\mathcal S (\mathbb R)$ as a Frechet space and $\mathcal S^* (\mathbb R)$ as a direct limit of Banach spaces.
Let $c:\mathcal S (\mathbb R) \otimes \mathcal S^* (\mathbb R) \to  \mathcal S^* (\mathbb R)$ be the convolution map. Let  $\hat c:\mathcal S (\mathbb R) \hat\otimes \mathcal S^* (\mathbb R) \to  \mathcal S^* (\mathbb R)$ be its extention to the completed tensor product. We have an argument that "proves" the following contradictory facts:

$\mathrm{Im} (c)=\mathrm{Im} (\hat c)$ 
$$\mathrm{Im} (c)=(f \in C^\infty(\mathbb R)|\exists \text{ a polinomial }p \text{ s.t. } \forall n\in \mathbb N \text{ the function } \frac{f^{(n)}}{p} \text{ is bounded}  )$$
$$\mathrm{Im} (\hat c)=(f \in C^\infty(\mathbb R)|\forall n\in \mathbb N, \exists \text{ a polinomial }p \text{ s.t. }  \text{ the function } \frac{f^{(n)}}{p} \text{ is bounded}  )$$
$\mathcal T_u(\mathbb R) \subsetneq \mathcal T(\mathbb R)$, were $\mathcal T_u(\mathbb R)$ is the r.h.s of (1) and $\mathcal T(\mathbb R)$ is the r.h.s of (2).

What of those statments are true and what are wrong? Do you have references for any of them?

REPLY [2 votes]: A bit late and far from answering all your questions but perhaps still worth mentioning:
A proof of the inclusion ${\rm Im}(c)\subseteq$ RHS of item 2) can be found in "Topological Vector Spaces and Distributions" by John Horváth. See in particular Proposition 4.11.7 on page 420.
BTW, as Johannes Hahn said the standard notation of for RHS of item 3) is $\mathcal{O}_M$. That for the RHS of item 2) is $\mathcal{O}_C$.