Convolution and space-time Fourier transform

[shekharc]

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

 

[jasonRF]

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.

would the relation G⊗(∂ⁿu/∂tⁿ) = ∂ⁿ(G⊗u)/∂tⁿ hold true?

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

 

[shekharc]

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.