Chapter 10 notes

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Chapter 9 notes

The history of the Hypercontractivity Theorem is complicated.

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Chapter 8 notes

The origins of the orthogonal decomposition described in Section 3 date back to the work of Hoeffding [Hoe48] (see also [vMis47]).

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Chapter 7 notes

The study of property testing was initiated by Rubinfeld and Sudan [RS96] and significantly expanded by Goldreich, Goldwasser, and Ron [GGR98]; the stricter notion of local testability was introduced (in the context of error-correcting codes) by Friedl and Sudan [FS95]. The first local tester for dictatorship was given by Bellare, Goldreich, and Sudan [BGS95,BGS98] (as [...]

Chapter 6 notes

The ${\mathbb F}_2$-polynomial representation of a boolean function $f$ is often called its algebraic normal form. It seems to have first been explicitly introduced by Zhegalkin in 1927 [Zhe27].

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Chapter 5 notes

Chow’s Theorem was proved by independently by Chow [Cho61] and by Tannenbaum [Tan61] in 1961; see also [Elg61].

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Chapter 4 notes

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Chapter 3 notes

The fact that the Fourier characters $\chi_{\gamma} : {\mathbb F}_2^n \to \{-1,1\}$ form a group isomorphic to ${\mathbb F}_2^n$ is not a coincidence; the analogous result holds for any finite abelian group and is a special case of the theory of Pontryagin duality in harmonic analysis. We will see further examples of this in Chapter [...]

Chapter 2 notes

The mathematical study of social choice began in earnest in the late 1940s; see Riker [Rik61] for an early survey or the compilation [BGR09] for some modern results.

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Chapter 1 notes

The Fourier expansion for real-valued boolean functions dates back to Walsh [Wal23] who introduced a complete orthonormal basis for $L^2([0,1])$ consisting of $\pm 1$-valued functions, constant on dyadic intervals.

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