O’Donnell’s conjecture

For a function $$\begin{gathered} f: \\ {-1,1 \\ }^n \\ to \\ -1,1 \end{gathered}$$ the relationship between the sum of its linear Fourier coefficients $hatf(i)$ (defined by $$\begin{gathered} hatf(i)≔ \\ frac12^n \\ su{m}_{x\in \\ {-1,1 \\ }^n}f(x)x_i \end{gathered}$$ for $$\begin{gathered} i=1,2, \\ ldots,n \end{gathered}$$ and …