Let $f\in L^1(\mathbb{R})$. Suppose that $\hat{f}$, the Fourier transform of $f$, is a positive function in $C_0(\mathbb{R})$. Does there exists any function $g\in L^1(\mathbb{R})$ with $|\hat{g}|^2=\hat{f}$?
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1$\begingroup$ so $f$ is the convolution of $g$ with itself; for example, if $f$ is a gaussian then $g$ is a gaussian with half the variance. $\endgroup$– Carlo BeenakkerCommented Feb 12, 2022 at 13:40
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$\begingroup$ In other words if $\hat{f}\geq0$, we are looking for the solution of the equation $f=g*g^* $ where $g^*(t)=\overline{g(-t)}$. Hold in general?! $\endgroup$– ABBCommented Feb 12, 2022 at 13:47
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$\begingroup$ related: math.stackexchange.com/questions/581143/… $\endgroup$– Onur OktayCommented Feb 15, 2022 at 17:42
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