Introduction to Analysis of Boolean Functions 1
summary
Develops Fourier analysis on finite abelian groups and the Boolean cube, proving that characters are roots of unity, that a non‑principal character sums to zero, and that characters form an orthonormal basis so f = Σχ f̂(χ)χ with coefficients f̂(χ)=⟨f,χ⟩. Shows the Fourier transform is unitary via the Parseval identity, and rewrites ±1-valued functions as unique multilinear polynomial expansions where degree = max |S| with f̂(S)≠0. Recommends splitting into low/high frequency levels and gives elementary proofs of orthogonality and the key lemma.
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