## §9.1: Low-degree polynomials are reasonable

As anyone who has worked in probability knows, a random variable can sometimes behave in rather “unreasonable” ways. It may be never close to its expectation. It might exceed its expectation almost always, or almost never. It might have finite $1$st, $2$nd, and $3$rd moments, but an infinite $4$th moment. All of this poor behaviour can cause a lot of trouble — wouldn’t it be nice to have a class of “reasonable” random variables?
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## Chapter 9: Basics of hypercontractivity

In 1970, Bonami proved the following central result:

The Hypercontractivity Theorem Let $f : \{-1,1\}^n \to {\mathbb R}$ and let ${1 \leq p \leq q \leq \infty}$. Then $\|\mathrm{T}_\rho f\|_q \leq \|f\|_p$ for $0 \leq \rho \leq \sqrt{\tfrac{p-1}{q-1}}$.

## Chapter 8 notes

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

## §8.6: Highlight: Randomized decision tree complexity

A decision tree $T$ for $f : \{-1,1\}^n \to \{-1,1\}$ can be thought of as a deterministic algorithm which, given adaptive query access to the bits of an unknown string $x \in \{-1,1\}^n$, outputs $f(x)$. E.g., to describe a natural decision tree for $f = \mathrm{Maj}_3$ in words: “Query $x_1$, then $x_2$. If they are equal, output their value; otherwise, query and output $x_3$.” For a worst-case input (one where $x_1 \neq x_2$) this algorithm has a cost of $3$, meaning it makes $3$ queries. The cost of the worst-case input is the depth of the decision tree.
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## §8.5: Abelian groups

The previous section covered the case of $f \in L^2(\Omega^n, \pi^{\otimes n})$ with $|\Omega| = 2$; there, we saw it could be helpful to look at explicit Fourier bases. When $|\Omega| \geq 3$ this is often not helpful, especially if the only “operation” on the domain is equality. For example, if $f : \{\mathsf{Red}, \mathsf{Green}, \mathsf{Blue}\}^n \to {\mathbb R}$ then it’s best to just work abstractly with the orthogonal decomposition. However if there is a notion of, say, “addition” in $\Omega$ then there is a natural, canonical Fourier basis for $L^2(\Omega, \pi)$ when $\pi$ is the uniform distribution.

## §8.4: $p$-biased analysis

Perhaps the most common generalized domain in analysis of boolean functions is the case of the hypercube with “biased” bits.
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## §8.3: Orthogonal decomposition

In this section we describe a basis-free kind of “Fourier expansion” for functions on general product domains. We will refer to it as the orthogonal decomposition of $f \in L^2(\Omega^n, \pi^{\otimes n})$ though it goes by several other names in the literature: e.g., Hoeffding, Efron–Stein, or ANOVA decomposition.
In this section we will revisit a number of combinatorial/probabilistic notions and show that for functions $f \in L^2(\Omega^n, \pi^{\otimes n})$, these notions have familiar Fourier formulas which don’t depend on the Fourier basis.