Chapter 8: Generalized domains

So far we have studied functions $f : \{0,1\}^n \to {\mathbb R}$. What about, say, $f : \{0,1,2\}^n \to {\mathbb R}$? In fact, very little of what we’ve done so far depends on the domain being $\{0,1\}^n$; what it has mostly depended on is our viewing the domain as a product probability distribution. Indeed, much of analysis of boolean functions carries over to the case of functions $f : \Omega_1 \times \cdots \times \Omega_n \to {\mathbb R}$ where the domain has a product probability distribution $\pi_1 \otimes \cdots \otimes \pi_n$. There are two main exceptions: the “derivative” operator $\mathrm{D}_i$ does not generalize to the case when $|\Omega_i| > 2$ (though the Laplacian operator $\mathrm{L}_i$ does); and, the important notion of hypercontractivity (introduced in the next chapter) depends strongly on the probability distributions $\pi_i$.

In this chapter we focus on the case where all the $\Omega_i$’s are the same, as are the $\pi_i$’s. (This is just to save on notation; it will be clear that everything we do holds in the more general setting.) Important classic cases include functions on the $p$-biased hypercube (Section 4) and functions on abelian groups (Section 5). For the issue of generalizing the range of functions — e.g., studying functions $f : \{0,1,2\}^n \to \{0,1,2\}$ — see the exercises.

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