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?
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- $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$Fabrice NEYRET– Fabrice NEYRET2016-03-24 09:59:52 +00:00Commented Mar 24, 2016 at 9:59
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1 Answer
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$f(x)=e^{|x|^2}$ 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 $\phi$.
