(What follows is motivated by an answer to https://mathoverflow.net/questions/433612/fourier-optimization-problem-related-to-the-prime-number-theorem?noredirect=1#comment1116702_433612)

Let $f:\mathbb{R}\to [0,\infty)$ be such that
<br> 
(a) $\int_{\mathbb{R}} f(x) dx = 1$,<br> 
(b) $\widehat{f}(t)=0$ for all real $t$ with $|t|>1$. <br>

What is the choice of $f$ such that $$\int_{\mathbb{R}} |x| f(x) dx$$is minimal? What is that minimum?

Remarks:
1. It is easy to see that we can assume $f$ to be an even function.
2. Yes, this seems to be yet another incarnation of the uncertainty principle.