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My chemist roommate asked me the following question. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a real-valued function and $F$ its Fourier transform. Suppose we know the modulus function $|F| : \mathbb{R} \rightarrow \mathbb{R}$. What can we deduce about $f$, can we determine it completely?

Feel free to assume any regularity conditions on $f$.

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  • $\begingroup$ If you translate $f$ you multiply $F$ by a function of absolute value $1$ so no. $\endgroup$
    – Torsten Ekedahl
    Commented Nov 1, 2011 at 5:46
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    $\begingroup$ We know it up to a certain group of transformations, including translations. More specifically, take any function of absolute value $1$ with odd imaginary part and even real part, and multiply the fourier transformation by that. This will create a translation-ish transformation of the function that the modulus of the fourier transform cannot detect. $\endgroup$
    – Will Sawin
    Commented Nov 1, 2011 at 6:32

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To try to determine a function from the absolute value of its Fourier transform is actually the famous "hidden phase problem". In X-ray crystollography one measures the absolute value of the Fourier transform of a function that describes where the atoms in the molecule are located. However, using clever tricks and some a priori knowledge of the unknown functions (for instance the fact that it is non-negative) one has been able to handle this problem in practice. The Nobel prize in chemistry 1985 was awarded for progress on this problem.

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  • $\begingroup$ Herbert Hauptman, who shared the 1985 Nobel prize in Chemistry for his work in X-ray crystallography, died on October 23. See this link for information on Hauptman and his work: ams.org/news?news_id=1289 $\endgroup$
    – Jan Boman
    Commented Nov 2, 2011 at 23:43
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See

http://www.optics.rochester.edu/workgroups/fienup/PUBLICATIONS/OL78_RecModFT.pdf

for a closely related question, and

Reconstruction of a function from the modulus of its Fourier transform V. V. Bashurov (math notes, 1969) for the exact question.

Your chemist friend is probably thinking of X-ray diffraction, where all you get is the modulus. There is an enormous body of work on this (usually the thing you are transforming has additional crystallographic symmetry.

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