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Analysis of Boolean functions studies real-valued functions on a Boolean hypercube using Fourier analysis, probability theory, and combinatorics.

The Bonami lemma states that a function of Fourier degree at most $k$ satisfies $\Vert f{\Vert }_4\le 3^{k/2}\Vert f{\Vert }_2$.

The noise operator averages $f(y)$ over a random $y$ correlated with $x$ by $\rho$. It acts diagonally on the Fourier-Walsh transform:

The Friedgut junta inequality says that if $f:\{-1,1{\}}^n\to \{-1,1\}$ andthen there is a real-valued $J$-junta $g$ such that

To prove it, put $J=\{i:{\mathrm{Inf}}_i(f)\ge \tau \}$. Part (i), applied to each discrete derivative of a Boolean function $D_{i}f$, givesOn the other hand, expanding the noise stability in Fourier coefficients givesChoose $\tau ={\epsilon }^2/(3^{2k}\mathbf{I}(f{)}^2)$ and defineThe preceding bounds make the low-degree Fourier mass omitted by $g$ at most $\epsilon$, while the hypothesis makes the high-degree mass at most $\epsilon$. Thus $\Vert f-g{\Vert }_2^2\le 2\epsilon$. Finally,which gives the asserted bound on $|J|$.

Solved by gpt-5.6-sol high.

Decompose $f={\sum }_{j=0}^n{f}^{(=j)}$ into its homogeneous Fourier levels. The Bonami lemma and the triangle inequality giveApply this estimate to the $m$-fold tensor power $f^{\otimes m}$. Tensor products multiply both relevant norms and commute with the noise operator on the Boolean hypercube, soTaking $m$th roots and the limit $m\to \infty$ proves the hypercontractive inequality on the Boolean hypercube

The noise operators are self-adjoint and satisfy $T_{\rho }{T}_{\sigma }=T_{\rho \sigma }$. By the duality of Lp spaces,Consequently

Solved by gpt-5.6-sol high.

Under the $p$-biased product measure ${\mu }_p$ on $\{-1,1{\}}^n$, the coordinates are independent random variables with $\mathbb{P}(x_i=-1)=p$ and $\mathbb{P}(x_i=1)=1-p$. Writing $\mu =1-2p$, $\sigma =2\sqrt{p(1-p)}$, and ${\varphi }_i=(x_i-\mu )/\sigma$, the products ${\varphi }_S={\prod }_{i\in S}{\varphi }_i$ form the $p$-biased Fourier basis.