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Fourier Transform of Distribution

  1. Jan 17, 2014 #1

    VVS

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    Hi,

    I hope somebody can help me with this one.

    1. The problem statement, all variables and given/known data
    Compute the Fourier Transform of the distribution x-a

    2. Relevant equations
    The Fourier Transform of a distribution is just the distribution evaluated with the Fourier Transform of a test function.


    3. The attempt at a solution
    See this pdf View attachment Übung 27.pdf
    I used integration by parts but now I am stuck, because I have to evaluate the integral of the Foureir Transform of the test function.


    Thanks
    VVS
     
  2. jcsd
  3. Jan 17, 2014 #2

    vanhees71

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    Science Advisor
    2016 Award

    I'd rather regularize the distribution, e.g., defining
    [tex]\tilde{f}_{\epsilon}(k)=\int_{\mathbb{R}} \mathrm{d} x \exp(-\mathrm{i} k x) (x-a) \exp(-\epsilon x^2).[/tex]
    Then you only need to know that
    [tex]\delta_{\epsilon}(k)=\frac{1}{2 \sqrt{\pi \epsilon}} \exp \left (-\frac{k^2}{4 \epsilon} \right )[/tex]
    is a smoothed [itex]\delta[/itex] distribution, i.e.,
    [tex]\lim_{\epsilon \rightarrow 0^+} \delta_{\epsilon}(k)=\delta(k).[/tex]
     
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