## §6.3: Constructions of various pseudorandom functions

In this section we give some constructions of boolean functions with strong pseudorandomness properties.

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## §5.4: Degree-1 weight

In this section we prove two theorems about the degree-$1$ Fourier weight of boolean functions: $\mathbf{W}^{1}[f] = \sum_{i=1}^n \widehat{f}(i)^2.$

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## §5.3: The Fourier coefficients of Majority

In this section we will analyze the Fourier coefficients of $\mathrm{Maj}_n$. In fact, we give an explicit formula for them in Theorem 16 below. But most of the time this formula is not too useful; instead, it’s better to understand the Fourier coefficients of $\mathrm{Maj}_n$ asymptotically as $n \to \infty$.

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## §5.2: Majority, and the Central Limit Theorem

Majority is one of the more important functions in boolean analysis and its study motivates the introduction of one of the more important tools: the Central Limit Theorem (CLT).

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## §5.1: Linear threshold functions and polynomial threshold functions

Recall from Chapter 2.1 that a linear threshold function (abbreviated LTF) is a boolean-valued function $f : \{-1,1\}^n \to \{-1,1\}$ that can be represented as $$\label{eqn:generic-LTF} f(x) = \mathrm{sgn}(a_0 + a_1 x_1 + \cdots + a_n x_n)$$ for some constants $a_0, a_1, \dots, a_n \in {\mathbb R}$.

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## §4.5: Highlight: LMN’s work on constant-depth circuits

Having derived strong results about the Fourier spectrum of small DNFs and CNFs, we will now extend to the case of constant-depth circuits. We begin by describing how Håstad applied his Switching Lemma to constant-depth circuits. We then describe some Fourier-theoretic consequences coming from a very early (1989) work in analysis of boolean functions [...]

## §4.4: Håstad’s Switching Lemma and the spectrum of DNFs

Let’s further investigate how random restrictions can simplify DNF formulas.

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