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$ u *v $ of two functions $ u,v ∈ L^1(Z(N)) $

I know the problem wants me to show that convolution is turned into multiplication but he is doing it in reference to set theory I think? I am not sure how to verify this.

He introduced set theory to try and teach fourier transforms. I have never had to do fourier transforms with set theory in this manor. I also have not done set theory in 3 years so I am struggling on the basic approach to this problem. Any help is appreciated.

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  • $\begingroup$ There is no set theory here. You are being asked to show that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. $\endgroup$ – spaceisdarkgreen Sep 17 '18 at 4:50
  • $\begingroup$ @spaceisdarkgreen How do I approach this? I have only ever had to do application based problems in my engineering classes so I am not sure how to approach this. $\endgroup$ – Mr. Dole Sep 17 '18 at 4:56
  • $\begingroup$ What is meant by $Z(N)$? $\endgroup$ – Bungo Sep 17 '18 at 4:58
  • $\begingroup$ @Bungo Group of Nth roots of unity $\endgroup$ – Mr. Dole Sep 17 '18 at 5:02
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I do not have the reputation to vote to close, but this question has already been answered here: Proof of the discrete Fourier transform of a discrete convolution.

To the OP: The vector notation $\langle a_0,a_1,\dots, a_{n-1}\rangle$ in that problem simply means the function that sends the identity element to $a_0$, the first root of unity $e^{2\pi i/n}$ to $a_1$, and so on until the last root $e^{2\pi i(n-1)/n}$ is sent to $a_{n-1}$.

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