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What is the relationship between the Fourier transform of an $L^1$ function $f: \mathbb{R}^d \to \mathbb{C}$ and the Fourier transform of $|f|$?

In other words, what is the relationship between $$ \widehat{f}(\xi) = \int e^{-2\pi i x \cdot \xi} f(x) dx $$ and $$ \widehat{\, | f |\, }(\xi) = \int e^{-2\pi i x \cdot \xi} |f(x)| dx? $$

Writing $f = g|f|$, we have $\widehat{f} = \widehat{g} \ast \widehat{\, | f |\, }$. But can we get something more explicit?

More generally, what is the relationship between the Fourier transform of a complex-valued finite measure $\mu$ and the Fourier transform of $|\mu|$, where $|\mu|$ is the the variation of $\mu$.

In other words, what is the relationship between $$ \widehat{\mu}(\xi) = \int e^{-2\pi i x \cdot \xi} d\mu(x) $$ and $$ \widehat{\, | \mu |\, }(\xi) = \int e^{-2\pi i x \cdot \xi} d|\mu|(x)? $$

Remark: The title of the following question is similar, but the content is different. Connection between the Fourier transform of f and |f|

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  • $\begingroup$ @Bombyxmori The OP already mentioned that question as being not particularly relevant. $\endgroup$ – Linden Oct 19 '17 at 2:16
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Not sure about $|f|,$ but for $g = |f|^2,$ the fourier transform of $g$ is given by $$\widehat{g(y)} = \frac{1}{2\pi}\int_{-\infty}^\infty \overline{F(t)} F(t + y) d t,$$ where $F$ is the fourier transform of $f.$ The above is apparently known as the Wiener-Khinchin(-Einstein) theorem.

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    $\begingroup$ Isn't that just $\widehat{|f|^2} = \widehat{ \, f \overline{f} \, } = \widehat{f} \ast \widehat{\overline{f}}$ ? $\endgroup$ – Linden Oct 19 '17 at 2:11
  • $\begingroup$ @Linden Something like that. Wiener and Khinchin actually cared about the convergence of this. Einstein (being a physicist) not so much. $\endgroup$ – Igor Rivin Oct 19 '17 at 2:31

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