Fourier transform of a real-valued function. 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 $f$.
 A: 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.
A: 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. 
