If $f$ has support on $[-x_0,x_0]$ can its Fourier transform $\tilde{f}$ be zero on $[-p_0,p_0]$? If so, what is the maximum admissible product $x_0p_0$?
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Let's assume that $f$ is not identically zero.
In this answer, it is shown that if $f$ has compact support, then $\hat{f}$ is entire. A non-zero entire function cannot be zero on a set with a limit point. Thus, if $f=0$ on $[-p_0,p_0]$, then $p_0=0$, and therefore, the maximum of $x_0p_0=0$.
