Recovering a function from limited info (Fourier transforms)

[pellman]

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.>

 

[DrClaude]

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 $|F(k)|^2$.

 

[pellman]

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.