I am having trouble with this question:
Show that there exists a compactly supported $C^\infty$ function $\phi$ on $\mathbf{R}$ such that $\phi \ge 0, \phi(0) >0$, and $\hat{\phi} \ge 0$.
I know that $\phi = e^{-\pi x^2}$ would work since $\hat{\phi} = e^{-\pi x^2}$ but this $\phi$ is not compactly supported...