ca.analysis and odes - Fourier transform of Analytic Functions

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

$f(x/mu)/mu=frac{2}{3} f(x) + frac{1}{3} (fast f)(x)quad$
where $ast$ mark the convolution, and $mu=frac{4}{3}$. $f$ is positive and real for $xin [0,infty)$

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.