# verify (u*v)^ = uv

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 4:50
• @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 4:56
• What is meant by $Z(N)$? – Bungo Sep 17 '18 at 4:58
• @Bungo Group of Nth roots of unity – Mr. Dole Sep 17 '18 at 5:02

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 $$a_{n-1}$$.