§8.5: Abelian groups

The previous section covered the case of $f \in L^2(\Omega^n, \pi^{\otimes n})$ with $|\Omega| = 2$; there, we saw it could be helpful to look at explicit Fourier bases. When $|\Omega| \geq 3$ this is often not helpful, especially if the only “operation” on the domain is equality. For example, if $f : \{\mathsf{Red}, [...]

§1.5: Probability densities and convolution

For variety’s sake, in this section we write the Hamming cube as ${\mathbb F}_2^n$ rather than $\{-1,1\}^n$. In developing the Fourier expansion, we have generalized from boolean-valued boolean functions $f : {\mathbb F}_2^n \to \{-1,1\}$ to real-valued boolean functions $f : {\mathbb F}_2^n \to {\mathbb R}$. Boolean-valued functions arise more often in [...]