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Can someone explain this step in the proof of the convolution theorem?

  1. Jun 17, 2011 #1
    I fail to understand a step made in this proof:
    http://en.wikipedia.org/wiki/Convolution_theorem" [Broken]

    more specifically the last step where the integral is written as a product of 2 seperate integrals (each equal to a fourier transform):
    from:
    30aa5ce6a4881f46515121f4ccfee81d.png
    to:
    97cc52195eb954a68ef235c3969e3f02.png
    I'm quite rusty on my integration, but as far I can remember this operation is only allowed when y is independent of x. (since y is taken out of an integral over x). But since we substituted y = z − x this is not the case. Can somebody explain me why this step is correct? (and why I'm wrong)
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Jun 17, 2011 #2
    After the substitution, y is just a dummy variable and consequently independent of x. If it was a definite integral, the substitution would have moved the dependency on x to the integral limits but because the integral is over all space, they are still independent of x after the substitution.
     
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