Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). If \(f:\{ 0,1\} ^n\to [0,1]\) and \(0\le \rho \le 1\), then \((T_\rho f)(x)\in [0,1]\) for every \(x\).

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For real \(c,d,\rho \) and real functions \(f,g\) on \(\{ 0,1\} ^n\), \(T_\rho (cf+dg)=cT_\rho f+dT_\rho g\).

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\), \(a_f=(f_0+f_1)/2\), and \(d_f=(f_0-f_1)/2\), and let \(s_0=1\), \(s_1=-1\). For every \(x,b\) and real \(\rho \), \((T_\rho f_b)(x)=(T_\rho a_f)(x)+s_b(T_\rho d_f)(x)\), where noise on the right acts on the first \(n\) coordinates.

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\), \(a_f=(f_0+f_1)/2\), and \(d_f=(f_0-f_1)/2\), and let \(s_0=1\), \(s_1=-1\). For every \(x,b\), \(D_n(T_\rho f)(x,b)=\rho (T_\rho d_f)(x)\), where \(D_n\) is the last-coordinate half-difference.

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\), \(a_f=(f_0+f_1)/2\), and \(d_f=(f_0-f_1)/2\), and let \(s_0=1\), \(s_1=-1\). For either bit \(b\), \((T_\rho f)_b=T_\rho a_f+s_b\rho T_\rho d_f\) as functions on \(\{ 0,1\} ^n\).

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\), \(a_f=(f_0+f_1)/2\), and \(d_f=(f_0-f_1)/2\), and let \(s_0=1\), \(s_1=-1\). For either bit \(b\), \(T_\rho f_b=T_\rho a_f+s_bT_\rho d_f\) as functions on \(\{ 0,1\} ^n\).

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\), \(a_f=(f_0+f_1)/2\), and \(d_f=(f_0-f_1)/2\), and let \(s_0=1\), \(s_1=-1\). If \(f\) takes values in \([0,1]\) and \(0\le \rho \le 1\), then \((T_\rho a_f)(x)\pm (T_\rho d_f)(x)\in [0,1]\) for every \(x\).