The discussion centers on the recovery of an unknown function f from its Fourier transform F, specifically when only the magnitudes |f(x)|² and |F(k)|² are known. It is established that one cannot uniquely recover f(x) due to counterexamples, such as f(x) = exp(-t²/2) and f(x) = i exp(-t²/2), which yield identical magnitudes. The modified question seeks to determine if f(x) can be recovered up to a constant factor of the form e^(iφ), which is relevant in quantum theory where such factors lack physical significance.
PREREQUISITESMathematicians, physicists, and students in advanced mathematics or quantum mechanics who are interested in Fourier analysis and function recovery techniques.
Thanks! So now I need to modify the question. Can we recover f(x) up to an overall constant factor of the form ##e^{i \phi}## ? The origin of the question is in quantum theory where such a constant factor has no physical significance.DrClaude said:Counter example: ##f(x) = \exp(-t^2/2)## and ##f(x) = i \exp(-t^2/2)## give the same results for ##|f(x)|^2## and ##|F(k)|^2##.