A version of the uncertainty principle says that a function and its Fourier transform cannot be both with compact support: it is not difficult to prove since a compactly supported distribution has an entire Fourier transform. Another version is that
$$
\left\Vert{\frac{du}{dx}}\right\Vert_{L^2(\mathbb R)}
\left\Vert{xu}\right\Vert_{L^2(\mathbb R)}\ge \frac12\left\Vert{u}\right\Vert_{L^2(\mathbb R)}^2,
$$
and many other quantitative versions are available for functions bounded above by Gaussians functions as well as their Fourier transform.

All this seems to be compatible with the existence of a function $u$ in $L^2(\mathbb R)$ such that
$$
\text{support u}\subset \mathbb R_+,\quad \text{support $\hat u$}\subset \mathbb R_+.
$$
My question: is there an "explicit" example of such a function?