The following basic result needs to be quoted on these matters of analyticity: the Paley-Wiener-Schwartz theorem gives a characterization of distributions with compact support. Let $u$ be a tempered distribution ; $u$ is compactly supported in a ball of center $0$ and radius $R$ if an only if $\hat u$ is an entire function such that $$\exists C_0, \exists N_0,\forall \zeta\in \mathbb C^n,\quad C^d,\quad \vert\hat u(\zeta)\vert\le C_0(1+\vert \zeta\vert)^{N_0}e^{R\vert\Im\zeta\vert}.$$ Something analogous allows a characterization of $C^\infty$ functions with compact support.
This leads to the following characterization of the analytic wave front set, due to Bros and Iagolnitzer. Let $v\in \mathcal E'(\mathbb R^d)$. We define the Fourier-Bros-Iagolnitzer transform $Tv$ of $v$ by the following formula, where the integral is in fact a bracket of duality, $$(Tv)(z,\lambda)=\int_{\mathbb R^d} e^{-\pi\lambda(z-x)^2} v(x) dx,\qquad z\in \mathbb C^d, \lambda >0.$$ Let $\Omega$ be an open subset of $\mathbb R^d$; let us note $\Omega\times(\mathbb R^d\backslash {0})$ by $\dot T^*(\Omega)$ and by $dL(z)$ the Lebesgue measure on $\mathbb C^d$. Let $u\in \mathcal D'(\Omega)$. The analytic wave-front-set of $u$, denoted by $WF_{A}(u)$, is the complement in $\dot T^*(\Omega)$ of the set of points $(x_{0},\xi_{0})$ such that $$\exists W_{0}\in \mathscr V_{x_{0}-i\xi_{0}}, \exists \chi_{0}\in C^\infty_c(\Omega), \chi_{0}(x)=1\ \text{near x_{0}}, \exists \epsilon_{0}>0\quad \text{with}$$ $$\sup_{\lambda\ge 1, z\in W_{0}}e^{\epsilon_{0}\lambda} \vert{(T\chi_{0} u)(z,\lambda)}\vert e^{-\pi\lambda(\Im z)^2} <+\infty.$$ The first projection of $WF_A(u)$ is the analytic singular support.
The following basic result needs to be quoted on these matters of analyticity: the Paley-Wiener-Schwartz theorem gives a characterization of distributions with compact support. Let $u$ be a tempered distribution ; $u$ is compactly supported in a ball of center $0$ and radius $R$ if an only if $\hat u$ is an entire function such that $$\exists C_0, \exists N_0,\forall \zeta\in \mathbb C^n,\quad \vert\hat u(\zeta)\vert\le C_0(1+\vert \zeta\vert)^{N_0}e^{R\vert\Im\zeta\vert}.$$ Something analogous allows a characterization of $C^\infty$ functions with compact support.