On the one-dimensional Boolean hypercube \(\{ 0,1\} ^1\), whose points are the functions from a single-element index set to \(\{ 0,1\} \), the set of all points consists of exactly the two constant functions: the one taking the value \(0\) (false) and the one taking the value \(1\) (true).

The power set of the singleton \(\{ 0\} \) is exactly \(\{ \varnothing ,\{ 0\} \} \).

The two functions from the singleton \(\{ 0\} \) to \(\{ 0,1\} \) that are constantly \(0\) and constantly \(1\) are distinct.

The empty subset \(\varnothing \) and the singleton subset \(\{ 0\} \) of the set \(\{ 0\} \) are distinct.

Let \(f:\{ 0,1\} ^1\to \mathbb {R}\) be a Boolean function of arity \(1\), and let \(x_0\in \{ 0,1\} ^1\) be the input whose single coordinate is \(\mathit{false}\) (that is, the sign encoding of that coordinate is \(+1\)). Then the value of \(f\) at this input equals the sum of its degree-zero and degree-one Fourier–Walsh coefficients:

Let \(f\colon \{ 0,1\} ^1\to \mathbb {R}\) be a function of a single bit, and write \(\hat f(\emptyset )\) and \(\hat f(\{ 0\} )\) for its two Fourier–Walsh coefficients, where \(\{ 0\} \) is the singleton frequency consisting of the sole coordinate. If \(x_1\) is the input whose single coordinate is \(1\) (the bit value corresponding to \(-1\) under the sign encoding), its value is the difference of these two coefficients,

Let \(f\colon \{ 0,1\} ^1\to \mathbb {R}\) be a Boolean function of arity one, and let \(\rho \) be any real number. Define \(T_\rho \) by multiplying the Fourier coefficient at each subset \(S\) of the coordinate set by \(\rho ^{|S|}\); then its expected square under the uniform measure is

where \(\emptyset \) and \(\{ 0\} \) are the only two frequencies for a single coordinate. 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 \(f\colon \{ 0,1\} ^1\to \mathbb {R}\) be a Boolean function of one bit and let \(p\in \mathbb {R}\). Writing \(\hat f(\emptyset )\) and \(\hat f(\{ 0\} )\) for its two Fourier–Walsh coefficients, the expectation of \(|f|^p\) under the uniform measure is

Here real powers use \(0^p=0\) for \(p\ne 0\) and \(0^0=1\).

Let \(n\ge 0\) be an integer, let \(f\colon \{ 0,1\} ^n\to \mathbb {R}\) be a function, and let \(r,s\) be real numbers with \(1\le r\le s\). Writing \(\mathbb {E}\) for the expectation under the uniform probability measure on \(\{ 0,1\} ^n\), the \(L^r\) norm of \(f\) is bounded by its \(L^s\) norm:

For every real number \(p\) with \(1 \le p \le 2\), one has \((p-1)^{p/2} \le 1\).

Let \(a, b, \rho \) be real numbers with \(0 \le \rho \le 1\). Then

For every real number \(x\), the product of its sign and itself equals its absolute value: \(\operatorname {sign}(x)\cdot x = \left\lvert x\right\rvert \), where \(\operatorname {sign}(x)\) is \(1\) if \(x\gt 0\), \(-1\) if \(x\lt 0\), and \(0\) if \(x=0\).

For every nonzero real number \(x\), the sign function \(\operatorname {sign}(x)\), equal to \(+1\) when \(x \gt 0\) and \(-1\) when \(x \lt 0\), satisfies \(\left\lvert \operatorname {sign}(x)\right\rvert = 1\).

Let \(n\ge 0\) be an integer, let \(f, g : \{ 0,1\} ^n \to \mathbb {R}\) be functions, and write \(\mathbb {E}[h] = 2^{-n}\sum _{x \in \{ 0,1\} ^n} h(x)\) for the expectation under the uniform measure and \(\langle f, g\rangle = \mathbb {E}[f\cdot g]\) for the \(L^2\) inner product. Then

Fix an integer dimension \(n\ge 0\) and real Hölder conjugate exponents \(p, q\) (so \(p, q \gt 1\) and \(\tfrac 1p + \tfrac 1q = 1\)). For a function \(g : \{ 0,1\} ^n \to \mathbb {R}\) write \(\lVert g\rVert _r = \bigl(\mathbb {E}\bigl[\, |g|^r\, \bigr]\bigr)^{1/r}\) for its \(L^r\) norm under the uniform measure on \(\{ 0,1\} ^n\). Then for every function \(u : \{ 0,1\} ^n \to \mathbb {R}\) there exists a function \(f : \{ 0,1\} ^n \to \mathbb {R}\) with \(\lVert f\rVert _p \le 1\) and \(\lVert u\rVert _q \le \langle f, u\rangle \), where \(\langle f, u\rangle = \mathbb {E}[f\cdot u]\).