bibkey: odonnell2014boolean authors: Ryan O’Donnell year: 2014 title: Analysis of Boolean Functions doi: null url: https://www.cs.cmu.edu/~odonnell/papers/Analysis-of-Boolean-Functions-by-Ryan-ODonnell.pdf claim: A function on the independent sign cube has a unique multilinear expansion; normalized parity characters give coefficient recovery and Parseval’s identity. strata_touched: [] license: citation-only triage: anchor

Analysis of Boolean Functions

The author-hosted PDF, chapter 1, Theorem 1.1 (printed p.22), Theorem 1.5 (§1.3), Proposition 1.8 and Parseval’s Theorem (printed p.25), supplies the real sign-cube identities

$$F(s)=\sum _{b\in \{0,1{\}}^d}{c}_b\prod _j{s}_j^{b_j},c_b=2^{-d}\sum _{s\in \{-1,1{\}}^d}F(s)\prod _j{s}_j^{b_j},$$

$$2^{-d}\sum _{s}F(s{)}^2=\sum _b{c}_b^2.$$

The normalization is uniform on all $2^d$ independent signs. The character orthogonality proof also gives the complex version with absolute squares, by applying the real identity to real and imaginary parts. The coordinate substitution $s_j=1-2t_j$ connects this unique multilinear polynomial to corner interpolation on $[0,1{]}^d$.

These ingredients are consumed in the guarded five-mode response bridge in §§1–2 of the seams continuation. A diagonal restriction $s_1=\cdots =s_d$ does not retain independent characters. Sign evaluations of a response polynomial are formal probes; the book supplies no quantum instrument, physical phase realization or probability law for the FIB source.