Explicit Schwartz $f$ such that $f = 1$ in $B(0, 1)$ and $\widehat{f} = 0$ outside $B(0, 1)$

Question

What is an explicit example of a Schwartz function $f$ such that $f = 1$ on the ball of radius 1 centered at the origin (in physical space) and $\widehat{f} = 0$ outside the ball of radius 1 centered at the origin (in frequency space)?

Answer 1

There is no such example, "explicit" or not.

You're assuming that $\hat f$ has compact support. It follows that $f$ is the restriction to $\Bbb R^n$ of an entire function in $\Bbb C^n$. In particular $f$ is real-analytic, and so if $f=1$ on the unit ball then $f=1$ everywhere.