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.
Best Answer
What is sufficient (though not necessary) is that the Fourier transform decays exponentially at $\infty$ (if you want just analyticity on the line) or faster than any exponent (if you want your original function to be entire). In particular, anything with compact support will do. If this is too restrictive for your construction, you'd better just tell what exactly you are trying to construct.