Recovering a function from limited info (Fourier transforms)

  • #1
675
4

Homework Statement


Consider some unknown function f:R --> C. Denote its Fourier transform by F. Suppose we know |f(x)|2 for all x and |F(k)|2 for all k. Can we recover f(x) (for all x) from this information?

Homework Equations


None.

The Attempt at a Solution


None. It's a yes or no question. Please just point me to the theorem if you know it. Thanks!

<Mentor note: approved.>
 
Last edited:

Answers and Replies

  • #2
DrClaude
Mentor
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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##.
 
  • #3
675
4
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##.
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.
 

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