I have the following problem: I have a function $f(x,t)$ with support on a compact region (red region figure). I am interested in what terms $ \widehat{\delta f} $ do I need to add to the Fourier transform $\widehat{f}$ such that the inverse Fourier transform of ${\widehat{f}+\widehat{\delta f}}$ has compact support on a sub region (blue region figure).
I would like to know what conditions do I need to require on $f$ such that this is possible. Are there any theorems related to this question? I know the Paley-Wiener theorem but this is kind of the inverse problem. Any directions to the solution would be greatly appreciated.