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Product of distributions

  1. Sep 7, 2009 #1
    I have a problem concerning the product of two tempered distributions.

    We have the temp. distr.
    [tex] F(x) = \dfrac{1}{4 \pi} \mathrm{sgn}(x^{0}) \delta(x^{2}) - \dfrac{i}{4 \pi} \mathcal{P}\dfrac{1}{x^{2}} [/tex],
    where x is a vector in Minkowski space and [tex]\mathcal{P}[/tex] is the principal value (in the sense of distributions).

    The Fourier transformed expression is:

    [tex] \hat{F}(p) = \dfrac{i}{2 \pi} \theta(p^{0}) \delta(p^{2}) [/tex].

    Now, my problem is that I don't know how to define the product [tex] F(x)^{2} [/tex]. I read in a book that this expression is well-definied, because if we consider the Fourier transform of this product, which will lead to a convolution
    [tex] (2 \pi)^{-2} \int d^{4} q \hat{F}(p-q) \hat{F}(q) [/tex]
    ,we obviously see that the intersection of the two [tex] \hat{F} [/tex] is compact.

    I don't "see" that this intersection is really compact. Do you know, why it is compact and why this "intersection-condition" is sufficient to prove that the product in coordinate-space [tex] F(x)^{2} [/tex] resp. the convolution is well-definied?
     
  2. jcsd
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