Fourier transform of a real-valued function. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:08:05Z http://mathoverflow.net/feeds/question/79681 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79681/fourier-transform-of-a-real-valued-function Fourier transform of a real-valued function. Jethro 2011-11-01T05:17:50Z 2011-11-01T10:15:38Z <p>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?</p> <p>Feel free to assume any regularity conditions on $f$.</p> http://mathoverflow.net/questions/79681/fourier-transform-of-a-real-valued-function/79688#79688 Answer by Igor Rivin for Fourier transform of a real-valued function. Igor Rivin 2011-11-01T08:23:23Z 2011-11-01T08:23:23Z <p>See </p> <p><a href="http://www.optics.rochester.edu/workgroups/fienup/PUBLICATIONS/OL78_RecModFT.pdf" rel="nofollow">http://www.optics.rochester.edu/workgroups/fienup/PUBLICATIONS/OL78_RecModFT.pdf</a></p> <p>for a closely related question, and</p> <p>Reconstruction of a function from the modulus of its Fourier transform V. V. Bashurov (math notes, 1969) for the exact question.</p> <p>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.</p> http://mathoverflow.net/questions/79681/fourier-transform-of-a-real-valued-function/79694#79694 Answer by Jan Boman for Fourier transform of a real-valued function. Jan Boman 2011-11-01T10:15:38Z 2011-11-01T10:15:38Z <p>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. </p>