restriction-exp]

Let $𝑓 : {\pm 1}^{𝑛} \to ℝ$ , $(𝐽, \overline{𝐽})$ be a partition of $[𝑛]$ , and fix $𝑆 \subseteq 𝐽$ . Then if we choose $𝑧 \sim {\pm 1}^{\overline{𝐽}}$ uniformly at random, we have

\begin{matrix} \underset{𝑧}{𝐄} [\hat{𝑓_{𝐽 | 𝑧}} (𝑆)] = \hat{𝑓} (𝑆) \\ \underset{𝑧}{𝐄} [\hat{𝑓_{𝐽 | 𝑧}} {(𝑆)}^{2}] = \sum_{𝑇 \subseteq \overline{𝐽}} \hat{𝑓} {(𝑆 \cup 𝑇)}^{2} . \end{matrix}

Proof. [#]

Both are statements about the Boolean function

𝐹_{𝑆 | \overline{𝐽}} 𝑓

{\pm 1}^{\overline{𝐽}}

, whose coefficients are

\hat{𝑓} (𝑆 \cup 𝑇)

by the note above. Its mean is its empty coefficient,

\hat{𝑓} (𝑆 \cup \emptyset) = \hat{𝑓} (𝑆)

, and its second moment is Parseval applied to it.