What is the Fourier Transform of an absolute function?

Question

I would like express that the Fourier transform of the function $$ |f(x)| $$ as $$ \widehat{|f|}(\xi) = \text{a function of } \widehat{f}(\xi) $$ In fact, I want to know the relation of $\widehat{|f|}(\xi)$ and $\widehat{f}(\xi)$. I think a identity $|f|^2 = f \overline{f}$ may help.

Answer 1

Let $f\in L^1$, $f\ge0$ and $a\in\mathbb{R}$. Define $f_a(x)=e^{iax}f(x)$. Then $$ \widehat{f_a}(\xi)=\hat f(\xi+a),\quad \widehat{|f_a|}(\xi)=\hat f(\xi). $$ There is no functional relation between the Fourier transforms of $f$ and $|f|$.