0

I have the following problem: I have a function $f(x,t)$ with support on a compact region (red region figure). I am interested in what terms $ \widehat{\delta f} $ do I need to add to the Fourier transform $\widehat{f}$ such that the inverse Fourier transform of ${\widehat{f}+\widehat{\delta f}}$ has compact support on a sub region (blue region figure).

Example regions

I would like to know what conditions do I need to require on $f$ such that this is possible. Are there any theorems related to this question? I know the Paley-Wiener theorem but this is kind of the inverse problem. Any directions to the solution would be greatly appreciated.

reuns
  • 77,999
Robert
  • 1
  • 1
  • I don't see how it is related to Paley-Wiener. What is supposed to be $\delta$ exactly ? And did you mean $\widehat{\delta f}$ or $\delta \widehat{ f}$ ? – reuns Mar 17 '17 at 17:54
  • $(\hat{f}+\widehat{\delta f})^{\vee} = f+ \delta f$ – abcd Mar 18 '17 at 07:34
  • $(\hat{f})^{\vee}=f$ – abcd Mar 18 '17 at 07:35
  • The relation to the Paley Wiener theorem is the following: I started with a function $f(x,t)$ having support everywhere. For non relevant reasons I am allowed to add arbitrary Fourier modes without changing the physical state of the function. The goal is to add Fourier modes to $\hat{f}$ in such a way that the resulting (physically the same) function becomes localized in a certain region. The Paley-Wiener theorem tells what conditions this function must obey such that the Fourier transform has compact support.

    With $\widehat{\delta f}$ I mean the Fourier modes I add.

     – Robert Mar 18 '17 at 10:27

0 Answers0