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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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can't you try to use a bump function that has a cut-off somewhere? –  Bombyx mori Sep 19 '12 at 4:21
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Take a compactly supported function and convolve it with itself; this will make the FT nonnegative. –  user31373 Sep 19 '12 at 4:34
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