Forgive me if this question is 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 $\mathcal{F}[f] (k)$?