For \(p\gt 0\) and \(f:\{ 0,1\} ^n\to \mathbb {R}\), the extended power mean agrees with the ordinary formula \(M_p(f)=(\mathbb {E}|f|^p)^{1/p}\) under the uniform measure.

For \(p\gt 0\) and every \(f:\{ 0,1\} ^n\to \mathbb {R}\), the uniform power mean \(M_p(f)=(\mathbb {E}|f|^p)^{1/p}\) is nonnegative.

For \(p\gt 0\) and functions \(f,g:\{ 0,1\} ^n\to \mathbb {R}\) satisfying \(0\le f(x)\le g(x)\) for every \(x\), one has \(M_p(f)\le M_p(g)\), where \(M_p(u)=(\mathbb {E}|u|^p)^{1/p}\).

Let \(p,c\gt 0\) and \(f:\{ 0,1\} ^n\to \mathbb {R}\). With \(M_p(u)=(\mathbb {E}|u|^p)^{1/p}\), one has \(M_p(cf)=cM_p(f)\).

Let \(p\gt 0\) and let \(f:\{ 0,1\} ^0\to [0,\infty )\). Its uniform power mean \(M_p(f)=(\mathbb {E}f^p)^{1/p}\) equals the value of \(f\) at the unique empty string.

Let \(p\gt 0\) and \(f:\{ 0,1\} ^{n+1}\to \mathbb {R}\), and write \(M_{p,z}\) for the uniform power mean \((\mathbb {E}_z|\cdot |^p)^{1/p}\). Then \(M_{p,(x,b)}f(x,b)=M_{p,x}\bigl(M_{p,b}f(x,b)\bigr)\), where \(x\in \{ 0,1\} ^n\) and \(b\in \{ 0,1\} \).

Let \(p\gt 0\) and \(F:\{ 0,1\} ^n\times \{ 0,1\} \to \mathbb {R}\). For uniform power means \(M_{p,z}u=(\mathbb {E}_z|u|^p)^{1/p}\), one has \(M_{p,x}(M_{p,b}F(x,b))=M_{p,b}(M_{p,x}F(x,b))\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\), \(c\in \mathbb {R}\), and \(S\subseteq [n]\), uniform Fourier–Walsh coefficients satisfy \(\widehat{cf}(S)=c\widehat f(S)\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) and \(\rho ,c\in \mathbb {R}\), the Fourier noise operator satisfies \(T_\rho (cf)=cT_\rho f\). 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.

Let \(\rho \in \mathbb {R}\) and \(f:\{ 0,1\} ^0\to \mathbb {R}\), the Fourier noise operator satisfies \(T_\rho f=f\). 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.

Let \(0\le \rho \le 1\) and \(f:\{ 0,1\} ^n\to [0,\infty )\). Then \(T_\rho f(x)\ge 0\) at every cube point \(x\). 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.

For \(\rho ,a\in \mathbb {R}\), let \(f(b)=1+a(-1)^b\) on \(\{ 0,1\} \). Then \(T_\rho f(b)=1+\rho a(-1)^b\) for both bits \(b\). 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.

Let \(f:\{ 0,1\} ^{n+1}\to \mathbb {R}\), \(f_b(x)=f(x,b)\), and \(g=(f_0+f_1)/2\). For every \(S\subseteq [n]\), \(\widehat g(S)=(\widehat{f_0}(S)+\widehat{f_1}(S))/2\).

Let \(f:\{ 0,1\} ^{n+1}\to \mathbb {R}\), \(f_b(x)=f(x,b)\), and \(h=(f_0-f_1)/2\). For every \(S\subseteq [n]\), \(\widehat h(S)=(\widehat{f_0}(S)-\widehat{f_1}(S))/2\).

Let \(f:\{ 0,1\} ^{n+1}\to \mathbb {R}\), \(\rho \in \mathbb {R}\), and \(f_b(z)=f(z,b)\). For \(x\in \{ 0,1\} ^n\) and \(b\in \{ 0,1\} \), \(T_\rho f(x,b)=T_\rho \bigl(t\mapsto T_\rho f_t(x)\bigr)(b)\); the outer operator acts on one bit and the inner operator on \(n\) bits. 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.