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.
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 $a_{n-1}$.