Having defined the basic operators of importance for functions on Gaussian space, it’s useful to also develop the analogue of the Fourier expansion.
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Having defined the basic operators of importance for functions on Gaussian space, it’s useful to also develop the analogue of the Fourier expansion. [...] 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}, [...] In this section we will revisit a number of combinatorial/probabilistic notions and show that for functions $f \in L^2(\Omega^n, \pi^{\otimes n})$, these notions have familiar Fourier formulas which don’t depend on the Fourier basis. [...] As we have seen, the Fourier expansion of $f : \{1,1\}^n \to {\mathbb R}$ can be thought of as the representation of $f$ over the orthonormal basis of parity functions $(\chi_S)_{S \subseteq [n]}$. In this basis, $f$ has $2^n$ “coordinates”, and these are precisely the Fourier coefficients of $f$. The “coordinate” of $f$ [...] 

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