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Hello,

Does anyone know a reference or proof of the "if" part of the following statement?

$$ \mu\in \mathcal{S}'(R)\quad\text{if and only if}\quad \mu*\alpha\in\mathcal{S}(R),\forall \alpha\in C_c^\infty(R), $$

where $\mathcal{S}'(R)$ is the space of Schwartz distributions, and $\mathcal{S}(R)$ is the space of smooth function with rapid decrease. I noticed that in Schwartz's book (1966) there is a proof. I am not good at french. I am wondering whether anyone knows any other references about this matter? It may use the Banach–Steinhaus theorem in certain way.

By the way, whether can we improve the above results by asking $\alpha\in\mathcal{S}(R)$?

Thank you very much for any hints. :-)

Anand

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You can choose $\alpha$ Schwartz distribution. Just apply the Fourier transform, which makes the convolution into a product. Since $C^\infty_c( \mathhb{R})$ is dense in $S(\mathbb{R})$, these statement should be equivalent. –  plusepsilon.de Jul 26 '11 at 20:49
    
About the first statement after Fourier transforming, you get a well defined functional $\widehat{\mu}: \widehat{ C_c^\infty(\mathbb{R})} \rightarrow \mathbb{C}$, and proving continuity you'd be done. –  plusepsilon.de Jul 26 '11 at 20:59
    
@pm, thank you. I see. How about the "if" part of the original statement? –  Anand Jul 26 '11 at 21:02
    
That is a suggestion for the "if" part, meaning that if it's well defined, then continuity comes for free... I mean this is basically the content of the statement, isn't? –  plusepsilon.de Jul 27 '11 at 5:43
    
Thanks pm. As for the if part, your hint is not directly obvious for me. I will have a careful consideration. :-) –  Anand Jul 30 '11 at 15:51
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1 Answer

"if" part can be explained in the following way: (it's only a sketch of the proof)

Step 1. Let us consider the Fréchet space F of all $C^{\infty}-$ functions with supports in $[-1, 1]$. It is easy to check that the linear operator A from F to $\mathcal{S}$ defined as $$ \alpha \mapsto \mu * \alpha $$ has a closed graphics and so by the well known theorem (for Fréchet spaces) is continuous.

Step 2. Then using basic properties of convolutions one can show that there exist integers n and C such that for each function $\varphi \in \mathcal{D}(\mathbb{R})$ we have $$ \sup{|\mu * \varphi|} \leq C \sup{|\varphi^{(n)}|} $$

Step 3. Now one can take any countable delta sequence and apply Fourier transform to prove(using obtained above estimate), that Fourier-images of convolutions of $\mu$ with members of the delta sequence converge in $\mathcal{S}'$ (it requires some additional work, but it seems that there are no principal difficulties here).

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