1
$\begingroup$

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...

  • 0
    can't you try to use a bump function that has a cut-off somewhere?2012-09-19
  • 4
    Take a compactly supported function and convolve it with itself; this will make the FT nonnegative.2012-09-19
  • 0
    Thanks! Actually is it more precise to convolve $f$ with $g(x):=\overline{f}(-x)$?2017-06-20

0 Answers 0