Is there is notion of Fourier transform of distribution?

Question

We note that every tempered distribution is a distribution.

Can we find a example of distribution which is NOT a tempered distribution?

Can we talk of Fourier transform of that distribution? If yes, how?

Answer 1

$f(x)=e^{|x|^2}$ is a distribution (since it is locally integrable) but is not a tempered distribution. Trying to define its Fourier transform the same way it is done for tempered distributions, that is, $$ \langle\hat f,\phi\rangle=\langle f,\hat\phi\rangle,\quad\phi\in\mathcal{D}(\mathbb{R}^n), $$ we run into problems: $\hat\phi\not\in\mathcal{D}(\mathbb{R}^n)$ and the right hand side is not defined for all $\phi$.