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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...

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    Thanks! Actually is it more precise to convolve $f$ with $g(x):=\overline{f}(-x)$?2017-06-20

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