derivative]

2026-09-08

For $𝑓 : {\pm 1}^{𝑛} \to ℝ$ and a coordinate $𝑖 \in [𝑛]$ , the (discrete) derivative of $𝑓$ in direction $𝑖$ is

𝐷_{𝑖} 𝑓 (𝑥) = \frac{𝑓 (𝑥^{𝑖 \to 1}) - 𝑓 (𝑥^{𝑖 \to - 1})}{2},

where $𝑥^{𝑖 \to 𝑏}$ is $𝑥$ with its $𝑖$ th coordinate set to $𝑏$ .

Proposition.Fourier Formula of the Derivative [#]

For $𝑓 : {\pm 1}^{𝑛} \to ℝ$ with expansion $𝑓 (𝑥) = \sum_{𝑆 \subseteq [𝑛]} \hat{𝑓} (𝑆) 𝜒_{𝑆} (𝑥)$ , the $𝑖$ th derivative of $𝑓$ is given by

𝐷_{𝑖} 𝑓 (𝑥) = \sum_{\begin{matrix} 𝑆 \subseteq [𝑛] \\ 𝑆 ∋ 𝑖 \end{matrix}} \hat{𝑓} (𝑆) 𝜒_{𝑆 ∖ {𝑖}} (𝑥) .