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 A is a set then I also use A(x) for the characteristic function of A, so that A(x)=1 when x\in A and 0 otherwise. Hopefully it will always be clear from context whether I am talking about the set or the function.

If G is a finite abelian group then we have the dual group of characters \widehat{G}, the set of homomorphisms from G to the circle group S^1. Given a function f:G\to\mathbb{C} we can define the Fourier transform \widehat{f}:\widehat{G}\to\mathbb{C} as

\widehat{f}(\gamma)=\mathbb{E}_{x\in G}f(x)\gamma(x).


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