§6.1: Notions of pseudorandomness

The most obvious spectral property of a truly random function $\boldsymbol{f} : \{-1,1\}^n \to \{-1,1\}$ is that all of its Fourier coefficients are very small (as we saw in Exercise 5.8).

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§1.5: Probability densities and convolution

For variety’s sake, in this section we write the Hamming cube as ${\mathbb F}_2^n$ rather than $\{-1,1\}^n$. In developing the Fourier expansion, we have generalized from boolean-valued boolean functions $f : {\mathbb F}_2^n \to \{-1,1\}$ to real-valued boolean functions $f : {\mathbb F}_2^n \to {\mathbb R}$. Boolean-valued functions arise more often in [...]