# What can be said about the Fourier expansion of compositions of Boolean functions, if anything?

Let $$\chi_S: \mathbb{F}_2^n \to \{\pm 1\}, x \mapsto (-1)^{\sum_{i \in S}x_i}$$ for every $$S \subseteq [n]$$ denote the parity functions from $$\mathbb{F}_2^n$$ to $$\{\pm 1\}$$. Then, of course, every $$f: \mathbb{F}_2^n \to \{\pm 1\}$$ can be written as a linear combination (over $$\mathbb{R}$$) of the parity functions. I have tried (without much ease or success), to produce the Fourier expansions of compositions of even simple Boolean functions, in particular of conjunctions of parity functions, like $$\chi_S \wedge \chi_T$$ (where under the encoding of $$1_{\mathbb{F}_2^n} \mapsto -1,0_{\mathbb{F}_2^n} \mapsto +1$$, $$\wedge$$ is just $$\max_2$$).

So, my questions are:

1. Are there good techniques/tools in general for computing/estimating the Fourier expansion of $$f \circ g$$ when we know both of the expansions for both $$f,g$$?
2. Is there something we can say about the particular case of conjunctions/disjunctions over parity functions?

If $$f,g$$ are two $$\pm1$$-valued functions and $$f \lor g$$ is their maximum (this is the standard notation for maximum; $$\land$$ stands for minimum), then $$f \lor g = \frac{1 + f + g - fg}{2}.$$ Since $$\chi_S \chi_T = \chi_{S \Delta T}$$, this shows that $$\chi_S \lor \chi_T = \frac{1}{2} + \frac{\chi_S}{2} + \frac{\chi_T}{2} - \frac{\chi_{S\Delta T}}{2}.$$ You can read the Fourier expansion from this (unless $$S = T$$, in which case $$\chi_S \lor \chi_S = \chi_S$$).