Suppose $f \in L^2(\mathbb{R})$ and let $X \subseteq \mathbb{R}$ be a set of finite measure. Here let $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Fourier transform.
Q: If the support of $\mathcal{F}(f)$ is contained in $X$, then is the support of $\mathcal{F}^{-1}(f)$ also contained in $X$?
I am not really sure how to even approach this problem. I feel like the answer is no, but cannot come up with any counter-examples. Perhaps this is a simple question, but I am not seeing it. Any help would be greatly appreciated.
Edit: From the comments below I know that we have $\mathcal{F}(f)(\xi) = \mathcal{F}^{-1}(f)(-\xi)$. After some playing around I think if $\mathcal{F}(f)$ is supported on $X$, then this implies that $\mathcal{F}^{-1}(f)$ is supported on $-X$ ($-X = \{-x: x \in X\}$). Is this correct?
Thanks! :)