## 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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## §1.6: Highlight: Almost linear functions and the BLR Test

In linear algebra there are two equivalent definitions of what it means for a function to be linear:

Definition 29 A function $f : {\mathbb F}_2^n \to {\mathbb F}_2$ is linear if either of the following equivalent conditions hold:

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