TITLE: Compactly supported function whose Fourier transform decays exponentially?
QUESTION [23 upvotes]: It's well known now that a function can not be compactly supported both on the space side and the frequency side (so-called uncertainty principle). On the other hand a function can have exponential decay on both sides, e.g. guassian function $e^{-x^2}$. My question is whether the intermediate case exists. More precisely, is there a function which is compactly supported on the space side and has exponential decay on the frequency side. Thanks!

REPLY [19 votes]: If $\left|\hat{f}(\xi)\right|\leqslant Ce^{-k|\xi|}$ , then by Fourier Inversion, we have
$$
\begin{align}
\left|f^{(n)}(x)\right|
&=\left|\int_{\mathbb{R}}\hat{f}(\xi)\,(i2\pi\xi)^n\,e^{i2\pi\,x\xi}\,\mathrm{d}\xi\right|\\
&\leqslant\int_{\mathbb{R}}Ce^{-k|\xi|}(2\pi|\xi|)^n\,\mathrm{d}\xi\\
&=(2C/k)(2\pi/k)^nn!\tag{1}
\end{align}
$$
Estimate $(1)$ implies the power series for $f$ has a radius of convergence of at least $\frac{k}{2\pi}$ everywhere. Thus, it is real analytic over all $\mathbb{R}$, and cannot have compact support unless it is identically $0$.