I will use a lot of notation without really explaining it, and some of it is not standard. I will list them here as I think of or use them. If there is something unexplained, please comment and I’ll add it on here.
If is a set then I also use for the characteristic function of , so that when and otherwise. Hopefully it will always be clear from context whether I am talking about the set or the function.
If is a finite abelian group then we have the dual group of characters , the set of homomorphisms from to the circle group . Given a function we can define the Fourier transform as