Fourier transform of a real-valued function. - MathOverflow

Fourier transform of a real-valued function.

2011-11-01T05:17:50Z

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 $f$.

Answer by Igor Rivin for Fourier transform of a real-valued function.

2011-11-01T08:23:23Z

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.

Answer by Jan Boman for Fourier transform of a real-valued function.

2011-11-01T10:15:38Z

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.