Manuscript I@2 · I.tex · section · 85631fa7

Signed slack representation

The independent static marginal problem › Signed slack representation · lines 2806–2829

For $T \subseteq S$ , define the Walsh character

χ_{T} (x) = i \in T \prod (- 1)^{x_{i}}

and normalized coefficient

f (T) = 2^{- ∣ S ∣} x \sum f (x) χ_{T} (x) .

Character orthogonality gives

2^{- ∣ S ∣} x \sum χ_{T} (x) χ_{U} (x) = {1, 0, T = U, T \neq = U,

and hence

f (x) = T \subseteq S \sum f (T) χ_{T} (x) .

Displayed formulas in this passage

unlabeled · line 2809
$χ_{T} (x) = i \in T \prod (- 1)^{x_{i}}$
unlabeled · line 2813
$f (T) = 2^{- ∣ S ∣} x \sum f (x) χ_{T} (x) .$
unlabeled · line 2817
$2^{- ∣ S ∣} x \sum χ_{T} (x) χ_{U} (x) = {1, 0, T = U, T \neq = U,$
unlabeled · line 2826
$f (x) = T \subseteq S \sum f (T) χ_{T} (x) .$