§11.2: Hermite polynomials

Having defined the basic operators of importance for functions on Gaussian space, it’s useful to also develop the analogue of the Fourier expansion.


§8.2: Generalized Fourier formulas

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.


§1.4: Basic Fourier formulas

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 $f$ [...]