On the Sum of Linear Coefficients of a Boolean Valued Function

(Here and everywhere else in the paper, the expectation E[·] is with respect to the uniform probability distribution on {−1, 1}.) The Parseval’s identity is the fact that ∑ S⊆[n] f̂(S) 2 = E[f(x)]. In particular, if f is boolean-valued then this implies that ∑ S⊆[n] f̂(S) 2 = 1. Given f : {−1, 1} → R and i ∈ [n], we define the discrete derivative ∂if : {−1, 1} → R by ∂if(x) = f(x1, x2, · · · , 1, xi+1, · · · , xn)− f(x1, x2, · · · ,−1, xi+1, · · · , xn) 2 . The influence of the ith coordinate on f is defined by Infi[f ] = E[(∂if) ] = ∑