Comp760, lectures 9-11: hypercontractivity, friedgut’s theorem, kkl inequality, chang’s lemma

Note that Tρ has a smoothing property. When ρ = 1, we have Tρf = f , but as one decreases ρ, the function Tρf “converges” to the constant E[f ] and indeed, for ρ = 0, we have Tρf = E[f ]. Note Tρf(x) takes the average of f evaluated at points sampled according to z. When ρ = 1, the random variable z is concentrated on point x, and thus the average is just over x so we obtain the original function f . As ρ decreases, the variable z becomes more spread out. Finally ρ = 0, we lose the information about x and z is distributed uniformly over all points in Z2 . Therefore in this case we get the constant function E[f ]. Recall from Lecture 3, when introducing the concept of convolution, we saw that if S is the Hamming ball of radius r around 0 in Z2 , then f ∗ 1S(x) is the