Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function, with $|f(x)| \le e^{-|x|}$ a.e.
Then how can we prove that its Fourier transform, $\hat{f}$, cannot have compact support (unless $f = 0$ a.e.).
I have a hint which says to show that $\hat{f} \in C^{\infty}$; I can do this using the differentiation rules for Fourier transforms, but am unsure of how to proceed from here. Can we use this to show that $\hat{f}$ is analytic in some neighbourhood of $\mathbb{R}$ (in which case the result follows easily)? I have another hint which says to then consider a suitable Taylor expansion. I am not too sure what to make of the second hint, but answers that involve the hints would be preferred.
You can show that $\hat{f}(s)$ is continuous in the strip $|\Im s| < 1$. Dominated convergence will do the job. Then you can use Morera's theorem to show that $\hat{f}$ is holomorphic in this strip by showing that $\int_{\Delta}\hat{f}(s)ds=0$ for every triangle in the same strip; all this requires is interchanging orders of integration.