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Show that Fourier transform does not violate causality

  1. Sep 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Show that Fourier transform does not violate causality. In other words, let [tex]\hat{E}(\omega)[/tex] be Fourier transform of function [tex]{E}(t)[/tex]. Show that [tex]E(t_1)[/tex], as evaluated from inverse fourier transform formula using [tex]\hat{E}(\omega)[/tex], does not depend on [tex]E(t_2)[/tex] for

    2. Relevant equations

    3. The attempt at a solution
    I don't quite get the meaning of ``does not depend''. Obviously, for sufficiently ``nice'' [tex]E(t)[/tex], the inverse fourier transform formula gives exactly [tex]E(t)[/tex] (if the transforms are appropriately normalized). So if one fixes the value [tex]E(t_1)[/tex] and perturbs [tex]E[/tex] for other [tex]t[/tex] (no matter in the past or in the future), the inverse fourier transform will still return the function which has value [tex]E(t_1)[/tex] at point [tex] t_1 [/tex].

    So, it looks like I don't understand the problem. Explanation is greatly appreciated.
  2. jcsd
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