Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function
$$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$
where $\sigma$ is a complex parameter. When $\Re (\sigma) \ge 0$, $g_\sigma$ is a tempered distribution$^{[1]}$ and so it is Fourier transformable.
On the contrary, it appears obvious that if $\Re(\sigma) <0$ then $g_\sigma$ is not tempered.
Question 1. What is the fastest way to prove this?
My guess is that one should exploit the fact that the pairing
$$\int_{\mathbb{R}^n} g_\sigma(x)\varphi(x)\, dx$$
makes no sense for some $\varphi \in \mathcal{S}(\mathbb{R}^n)$. But is it enough? I am afraid that this argument is incomplete.
Question 2. More generally, is there some characterization of tempered functions, that is, functions which belong to the space $L^1_{\text{loc}}(\mathbb{R})\cap \mathcal{S}'(\mathbb{R})$?
The only tempered functions that I know are polynomially growing functions. By this I mean the functions of the form $Pu$, where $P$ is a polynomial and $u \in L^p(\mathbb{R}^n)$ for some $p\in[1, \infty]$.
Question 3. Is it true that all tempered functions are polynomially growing functions?
$^{[1]}$ The definition of tempered distribution I refer to is the following.
A distribution $T \in \mathcal{D}'(\mathbb{R}^n)$ is called tempered if for every sequence $\varphi_n \in \mathcal{D}(\mathbb{R}^n)$ such that $\varphi_n \to 0$ in the Schwartz class sense, it happens that $\langle T, \varphi_n \rangle \to 0$. If this is the case then $T$ uniquely extends to a continuous linear functional on $\mathcal{S}(\mathbb{R}^n)$ and we write $T \in \mathcal{S}'(\mathbb{R}^n)$.
Best Answer
Question 1 What you have is almost enough. Assume $\Re\sigma \leq -\epsilon < 0$. Test $\exp (-\sigma |x|^2)$ "against" $\phi(x) = \exp( (\sigma+\epsilon/2)|x|^2)$ in the following way: you can construct a sequence of annular cut-off functions $\chi_k$ such that $\chi_k \phi \to 0$ in $\mathcal{S}$ (using the exponential decay of $\phi$) and $\langle g_\sigma, \chi_k\phi\rangle > c > 0$ for all $k$.
Question 2 You have the structure theorem of tempered distributions. (See Theorem 8.3.1 in Friedlander and Joshi).
Theorem Every tempered distribution is a (distributional) derivative of finite order of some continuous function of polynomial growth.
If you intersect against $L^1_{loc}$, this just guarantees that the distributional derivative is actually the weak derivative. From this you can conclude that an appropriate version of what you stated is true.