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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