 Fourier-Walsh transform

For $f:\{-1,1{\}}^n\to \mathbb{R}$, the characters ${\chi }_S(x)={\prod }_{i\in S}{x}_i$ form an orthonormal basis, and the Fourier-Walsh expansion is

On the finite Boolean hypercube, these coefficients are the Hadamard transform of the vector of function values, with expectation normalization.