On decompositions of multivariate functions

We present formulas that allow us to decompose a function f of d variables into a sum of 2ᵈ terms fᵤ indexed by subsets u of \1,… ,d\, where each term fᵤ depends only on the variables with indices in u. The decomposition depends on the choice of d commuting projections ⱼ\ⱼ₌₁ᵈ, where Pⱼ(f) does not depend on the variable xⱼ. We present an explicit formula for fᵤ, which is new even for the