Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(P(x)=\sum _{S\subseteq [n]}a_S\chi _S(x)\) with real coefficients \(a_S\). For every \(S\subseteq [n]\), \(\widehat P(S)=a_S\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(P(x)=\sum _{S\subseteq [n]}a_S\chi _S(x)\) with real coefficients \(a_S\). If \(k\ge 0\) is an integer and \(a_S=0\) whenever \(|S|\gt k\), then \(\widehat P(S)\ne 0\) implies \(|S|\le k\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(P(x)=\sum _{S\subseteq [n]}a_S\chi _S(x)\) with real coefficients \(a_S\). Then \(\mathbb E[P(x)P(x)]=\sum _{S\subseteq [n]}a_S^2\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(P(x)=\sum _{S\subseteq [n]}a_S\chi _S(x)\) with real coefficients \(a_S\). The second moment satisfies \(\mathbb E[P^2]=\sum _{S\subseteq [n]}a_S^2\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(P(x)=\sum _{S\subseteq [n]}a_S\chi _S(x)\) with real coefficients \(a_S\). For every real cube function \(u\), \(\mathbb E[Pu]=\sum _{S\subseteq [n]}a_S\widehat u(S)\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(P(x)=\sum _{S\subseteq [n]}a_S\chi _S(x)\) with real coefficients \(a_S\). For every \(S\subseteq [n]\), \(|a_S|\le \mathbb E|P|\).

Let \(n\ge 0\) and let \((a_S)_{S\subseteq [n]}\) be real coefficients that are not all zero. Then \(\sum _{S\subseteq [n]}a_S^2\gt 0\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(P(x)=\sum _{S\subseteq [n]}a_S\chi _S(x)\) with real coefficients \(a_S\). If \(f(x)=1\) when \(P(x)\ge 0\) and \(f(x)=-1\) when \(P(x)\lt 0\), then \(\mathbb E|P|=\mathbb E[Pf]\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(P(x)=\sum _{S\subseteq [n]}a_S\chi _S(x)\) with real coefficients \(a_S\). If \(a_S=0\) for \(|S|\gt 1\), then \(P(x)=a_\varnothing +\sum _{i=1}^n a_{\{ i\} }(-1)^{x_i}\) for every \(x\in \{ 0,1\} ^n\).