FKN, first proof, rewritten

Abstract

About twenty years ago we wrote a paper, "Boolean Functions whose Fourier Transform is Concentrated on the First Two Levels", \cite{FKN}. In it we offered several proofs of the statement that Boolean functions $f(x_1,x_2,\dots ,x_n)$, whose Fourier coefficients are concentrated on the lowest two levels are close to a constant function or to a function of the form $f=x_k$ or $f=1-x_k$. Returning to the paper lately, we noticed that the presentation of the first proof is rather cumbersome, and includes several typos. In this note we rewrite that proof, as a service to the public.