Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), \(x\in \Omega \), and \(S\subseteq [n]\). The Walsh coefficient at \(S\) of \(r\mapsto R_rf(x)=\sum _T\chi _T(r)f_T(x)\) is \(f_S(x)\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube and \(k\in \mathbb N\). The Fourier projection \(f^{\le k}\) has Fourier degree at most \(k\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube and \(k\in \mathbb N\). Then \(\| f^{\le k}\| _2\le \| f\| _2\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube, \(k\in \mathbb N\), \(q\ge 2\), and \(0\lt \rho \le 1\). Then \(\| f^{\le k}\| _q\le (\sqrt{q-1}/\rho )^k\| T_\rho f\| _q\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(q\gt 0\), \(a\ge 0\), and \(f:\Omega \to \mathbb R\). Then \(\| af\| _{q,P}=a\| f\| _{q,P}\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f,g:\Omega \to \mathbb R\), \(q\gt 0\), and \(A\ge 0\). If \(\| r\mapsto R_rf(x)\| _q\le A\| r\mapsto R_rg(x)\| _q\) on the uniform sign cube for every \(x\), then \(\| Rf\| _q\le A\| Rg\| _q\), where \(\| Rh\| _q=(\mathbb E_{x\sim P}\mathbb E_r|R_rh(x)|^q)^{1/q}\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(S\subseteq [n]\), \(k\in \mathbb N\), and \(f:\Omega \to \mathbb R\). The \(S\)-component of \(f^{\le k}\) equals \(f_S\) if \(|S|\le k\), and zero otherwise.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(\rho \in \mathbb R\), \(f:\Omega \to \mathbb R\), and \(x\in \Omega \). Then \(r\mapsto R_r(T_\rho f)(x)\) equals \(T_\rho (r\mapsto R_rf(x))\), with the noise on the right acting on the uniform Boolean sign cube.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(k\in \mathbb N\), \(f:\Omega \to \mathbb R\), and \(x\in \Omega \). Then \(r\mapsto R_r(f^{\le k})(x)\) is the Fourier projection to levels at most \(k\) of the function \(r\mapsto R_rf(x)\) on the uniform Boolean sign cube.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(k\in \mathbb N\), and \(f,g:\Omega \to \mathbb R\). Then \(\mathbb E_P[f^{\le k}g]=\mathbb E_P[fg^{\le k}]\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(\rho \in \mathbb R\), \(k\in \mathbb N\), and \(f:\Omega \to \mathbb R\). Then \(T_\rho (f^{\le k})=(T_\rho f)^{\le k}\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(q\gt 1\), \(A\gt 0\), and an operator \(T\) on real functions on \(\Omega \). If \(\mathbb E_P[(Tf)g]=\mathbb E_P[fTg]\) for every \(f,g\) and \(\| Tg\| _{q/(q-1),P}\le A\| g\| _{q/(q-1),P}\) for every \(g\), then \(\| Tf\| _{q,P}\le A\| f\| _{q,P}\) for every \(f\).