What functions can be obtained as a convolution of a Schwartz function and a tempered distribution? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:07:16Z http://mathoverflow.net/feeds/question/101763 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101763/what-functions-can-be-obtained-as-a-convolution-of-a-schwartz-function-and-a-temp What functions can be obtained as a convolution of a Schwartz function and a tempered distribution? Rami 2012-07-09T11:45:25Z 2012-08-06T17:22:01Z <p>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.</p> <p>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:</p> <ol> <li>$\mathrm{Im} (c)=\mathrm{Im} (\hat c)$ </li> <li>$$\mathrm{Im} (c)=(f \in C^\infty(\mathbb R)|\exists \text{ a polinomial }p \text{ s.t. } \forall n\in \mathbb N \text{ the function } \frac{f^{(n)}}{p} \text{ is bounded} )$$</li> <li>$$\mathrm{Im} (\hat c)=(f \in C^\infty(\mathbb R)|\forall n\in \mathbb N, \exists \text{ a polinomial }p \text{ s.t. } \text{ the function } \frac{f^{(n)}}{p} \text{ is bounded} )$$</li> <li>$\mathcal T_u(\mathbb R) \subsetneq \mathcal T(\mathbb R)$, 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).</li> </ol> <p>What of those statments are true and what are wrong? Do you have references for any of them?</p> http://mathoverflow.net/questions/101763/what-functions-can-be-obtained-as-a-convolution-of-a-schwartz-function-and-a-temp/101782#101782 Answer by Bazin for What functions can be obtained as a convolution of a Schwartz function and a tempered distribution? Bazin 2012-07-09T15:41:38Z 2012-07-09T15:41:38Z <p>The answer to the question in the title is: {Fourier $(\phi u)$} $_{\phi\in \mathscr S, u\in \mathscr S'}$.</p>