Let \(n\) be a natural number and let \(\rho \) be real with \(0\le \rho \le 1\). For every \(x,y\in \{ 0,1\} ^n\),

where \(\sigma (0)=1\) and \(\sigma (1)=-1\).

For every natural number \(n\), every real number \(\rho \) and all points \(x, y\) of the Boolean hypercube \(\{ 0,1\} ^n\),

where the sum ranges over all subsets \(S\) of \([n]=\{ 1,\dots ,n\} \), \(\chi _S\) is the Walsh–Fourier character indexed by \(S\), and \(\sigma \) is the sign encoding sending \(0\mapsto 1\) and \(1\mapsto -1\).

Fix \(\rho \in \mathbb {R}\), an arity \(n\), and a point \(x \in \{ 0,1\} ^n\). For every Boolean function \(g : \{ 0,1\} ^n \to \mathbb {R}\), the value of the noise operator \(T_\rho \) at \(x\) equals the kernel-weighted sum of \(g\), namely

where \(K_\rho (x,y) = \prod _{i} \tfrac {1 + \rho \, \sigma (x_i)\, \sigma (y_i)}{2}\) is the noise kernel and \(\sigma \) is the \(\pm 1\) sign encoding of a bit. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Fix a dimension \(n\) and a real number \(\rho \). For any two Boolean functions \(f, g : \{ 0,1\} ^n \to \mathbb {R}\), the \(L^2\) inner product of \(f\) with the noised function \(T_\rho g\) can be written as a kernel-weighted double sum:

where \(K_\rho (x,y) = \prod _{i} \tfrac {1 + \rho \, \sigma (x_i)\, \sigma (y_i)}{2}\) is the noise kernel and \(\sigma \) is the \(\pm 1\) sign encoding of a bit. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Fix a dimension \(n\) and a correlation parameter \(\rho \) with \(0 \le \rho \le 1\), and let \(K_\rho (x,y) = \prod _{i} \tfrac {1 + \rho \, \chi (x_i)\, \chi (y_i)}{2}\) denote the noise kernel on the Boolean hypercube \(\{ 0,1\} ^n\), where \(\chi (0)=1\) and \(\chi (1)=-1\). If \(h, h' : \{ 0,1\} ^n \times \{ 0,1\} ^n \to \mathbb {R}\) satisfy \(h(x,y) \le h'(x,y)\) for all \(x\) and \(y\), then the kernel-weighted average of \(h\) is at most that of \(h'\):

Fix a natural number \(n\), \(\rho \in \mathbb {R}\), and points \(x', y'\) in the Boolean hypercube \(\{ 0,1\} ^n\), and bits \(b, b'\). Write \((x', b)\) and \((y', b')\) for the points of \(\{ 0,1\} ^{n+1}\) obtained by appending \(b\) and \(b'\), respectively, as the final coordinate. Then the noise kernel of the extended pair factors as

where \(\chi (0)=1\) and \(\chi (1)=-1\).

Let \(n\) be a natural number and let \(h : \{ 0,1\} ^{n+1} \to \mathbb {R}\) be a Boolean function on \(n+1\) bits. Writing \((x', b) \in \{ 0,1\} ^n \times \{ 0,1\} \) for the point of \(\{ 0,1\} ^{n+1}\) obtained by appending the bit \(b\) to \(x'\) as its last coordinate, the uniform expectation of \(h\) can be computed by first averaging over the last bit and then over the remaining \(n\) bits:

Let \(n\) be a natural number, let \(p \gt 0\) and let \(f : \{ 0,1\} ^{n+1} \to \mathbb {R}\) be a Boolean function of arity \(n+1\). Writing \((x', b) \in \{ 0,1\} ^n \times \{ 0,1\} \) for the point of \(\{ 0,1\} ^{n+1}\) whose first \(n\) coordinates are \(x'\) and whose last coordinate is \(b\), one has

where both expectations are taken with respect to the uniform measure on the respective hypercubes.

Fix a natural number \(n\), a real number \(\rho \), and a function \(F \colon \{ 0,1\} ^{n+1} \times \{ 0,1\} ^{n+1} \to \mathbb {R}\) on pairs of points of the Boolean hypercube of dimension \(n+1\). Write \(\chi (0)=1\) and \(\chi (1)=-1\) for the sign encoding of a bit, \(K_\rho (x,y)=\prod _i\frac{1+\rho \chi (x_i)\chi (y_i)}{2}\) for the noise kernel, and \((x',b)\) for the point of \(\{ 0,1\} ^{n+1}\) obtained by appending the bit \(b\) to \(x' \in \{ 0,1\} ^n\). Then

where \(x,y\) range over \(\{ 0,1\} ^{n+1}\), the points \(x',y'\) range over \(\{ 0,1\} ^{n}\), and the bits \(b,b'\) range over \(\{ 0,1\} \).

Fix a natural number \(n\), a real number \(\rho \), Boolean functions \(f, g : \{ 0,1\} ^{n+1} \to \mathbb {R}\), and points \(x', y' \in \{ 0,1\} ^n\). For a bit \(b\), write \((x', b)\) for the point of \(\{ 0,1\} ^{n+1}\) obtained by appending \(b\) as the last coordinate of \(x'\), and let \(\sigma \) denote the sign encoding with \(\sigma (0)=1\) and \(\sigma (1)=-1\). Then

where \(F, G : \{ 0,1\} ^1 \to \mathbb {R}\) are the one-bit slices \(F(t) = f(x', t_0)\) and \(G(t) = g(y', t_0)\), and the inner product and the noise operator \(T_\rho \) are taken on the one-dimensional cube \(\{ 0,1\} ^1\). For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Fix a natural number \(n\) and \(p \gt 0\), let \(f : \{ 0,1\} ^{n+1} \to \mathbb {R}\) be a Boolean function, and let \(x' \in \{ 0,1\} ^n\). Writing \((x', b) \in \{ 0,1\} ^{n+1}\) for the point obtained by appending a bit \(b \in \{ 0,1\} \) to \(x'\), and taking the expectation over the single free bit under the uniform measure, one has

Let \(n\) be a natural number, let \(p \ge 1\) and let \(f : \{ 0,1\} ^{n+1} \to \mathbb {R}\) be a Boolean function on the \((n+1)\)-dimensional cube. For a point \(x' \in \{ 0,1\} ^n\), write \((x',0)\) and \((x',1)\) for the two extensions of \(x'\) obtained by appending a final bit. Then averaging over the last coordinate leaves the expected \(p\)-th power of \(\left\lvert f\right\rvert \) unchanged:

where both expectations are taken with respect to the uniform measure.