restriction]

2026-09-22

Let $𝑓 : {\pm 1}^{𝑛} \to ℝ$ and take $(𝐽, \overline{𝐽})$ to be a partition of $[𝑛]$ . Take $𝑧 \in {\pm 1}^{\overline{𝐽}}$ . Then the restriction of $𝑓$ to $𝐽$ using $𝑧$ is

\begin{matrix} 𝑓_{𝐽 | 𝑧} : {\pm 1}^{𝐽} \to ℝ \\ 𝑦 \mapsto 𝑓 (𝑦, 𝑧), \end{matrix}

the subfunction of $𝑓$ given by fixing the coordinates in $\overline{𝐽}$ to $𝑧$ .

When the partition $(𝐽, \overline{𝐽})$ is clear, we will often write the restriction as $𝑓_{| 𝑧}$ .

Warning. [#]

Even though we write

𝑓_{𝐽 | 𝑧} (𝑦) = 𝑓 (𝑦, 𝑧),

remember that we (generally) are not working with the literal concatenation of $𝑦$ and $𝑧$ .