TITLE: Fourier transform of Analytic Functions
QUESTION [12 upvotes]: Forgive me if this question does not meet the bar for this forum. But i would really appreciated some help.
I'm trying to construct a function according to some conditions in the frequency domain of the Fourier transformation. I want the function to be analytic and real when I transform it back to the time domain. The Fourier transformation of $f$ has of course some symmetry criteria to make $f$ real. But what about the Analytic property. As an analytic function imply some convergent power series expansion, and the Fourier transform of a polynomial is a sum of derivatives of Delta functions, I assume that there is a corresponding criteria of the Fourier transformation.
So the question is:
If a function $f:\mathbb{R}\rightarrow \mathbb{R}$ is assumed to be analytic, what is the corresponding criteria for the Fourier transform of the function $\mathcal{F}[f] (k)$?
Edit: what I am trying to construct is probability distribution with the following condition
$f(x/\mu)/\mu=\frac{2}{3} f(x) + \frac{1}{3} (f\ast f)(x)\quad$
where $\ast$ mark the convolution, and $\mu=\frac{4}{3}$. $f$ is positive and real for $x\in [0,\infty)$
Taking the fourier transformation make the condition simpler:
$\tilde f(\mu k) = \frac{2}{3}\tilde f(k) + \frac{1}{3}\tilde f^2(k)$
So my problem is to construct $f$ (I am in particular interested in the tail behavior) and I try to use the properties of $\tilde f$. I posted a similar problem a while ago (see here). Julián Aguirre answered how to construct $\tilde f$ if it is analytic. But the inverse transformation of the power expansion is an infinite sum of derivatives of Delta functions, and is of little help.

REPLY [4 votes]: 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
$$\exists C_0, \exists N_0,\forall \zeta\in \mathbb C^d,\quad
\vert\hat u(\zeta)\vert\le C_0(1+\vert \zeta\vert)^{N_0}e^{R\vert\Im\zeta\vert}.
$$
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.