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?

nobodyis actively following these tags, so nobody is going to see this question highlighted in their favorite tags. Not to mention, all three tags are terrible. I don't know enough to tell you anything about the question, or about the correct tags to use, but you might want to look at the Tags page and find a better tag for this question, so it will get better exposure. $\endgroup$