verify (u*v)^ = uv

Question

Picture included because I could not format correctly

$ 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.

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. — spaceisdarkgreen, Sep 17 '18 at 04:50
@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. — Mr. Dole, Sep 17 '18 at 04:56

score 1 · Answer 1 · answered Oct 10 '18 at 04:49

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 $a_{n-1}$.

verify (u*v)^ = uv

1 Answers1