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?
Best Answer
The answer to the question in the title is: {Fourier $(\phi u)$} $_{\phi\in \mathscr S, u\in \mathscr S'}$.