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Convolution and space-time Fourier transform

  1. Aug 18, 2015 #1
    Hi,

    I have a general function u(x,y,z,t). Then, (1) what would be the space-time Fourier transform of G⊗(∂nu/∂tn) and (2) would the relation G⊗(∂nu/∂tn) = ∂n(G⊗u)/∂tn hold true? Here, note that the symbol ⊗ represents convolution and G is a function of (x,y,z) only (i.e. it does not depend on time).

    Any answer would appreciated. Thanks!!

    -Chandra
     
  2. jcsd
  3. Aug 21, 2015 #2

    jasonRF

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    Gold Member

    Chandra,

    Most of your question is answered by applying the properties of Fourier transforms, all of which are, for exmaple, at:
    http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html
    including the convolution theorem (in link above and in this link: https://en.wikipedia.org/wiki/Convolution_theorem )
    and a knowledge of the Fourier transform of a derivative.
    Here you are essentially asking a question about interchanging differentiation and integration. As an engineer, I typically deal with "nice" functions for which this holds (indeed, I assume it holds!), but it doesn't always hold for any choice of functions. I cannot help you much more than that - sorry!
    jason
     
  4. Aug 23, 2015 #3
    Dear Jason,

    Thanks for your suggestions. In fact, I was a bit confused because of involvement of both space and time in the Fourier transform. Anyway, I did it (hopefully correctly) by taking Fourier transforms two times; first, I took the transform with respect to space, and then with respect to time. As for the 2nd question, "u" is not a very nice function--it does not converge to zero when x,y,z-->INFINITY. So, currently I am looking at whether I can use generalized Fourier Transforms to deal with it.
     
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