The article talks about composing these shapes using variable coefficients to fit the resulting emotional arcs. I'm not convinced this isn't just Fourier decomposition on arbitrary continuous functions.
Maybe there's insight there regardless? Hard to say. But the title could easily be "continuous functions are approximated by Fourier series with six terms" in my opinion.
In math, the obvious things aren't always true and the true things are often not obvious.
Trivially, the identity f(x) = x satisfies the guarantee as well. What amounts to insightful observation is the definition and classification of these functions. In exploring their existence in various forms, we can begin to understand what properties these functions share.
So the interesting part is not that this class of function _exists_, because of course it does! Your intuition has led you to three possible candidates. But if we limit ourselves to only the functions that satisfy the condition _wave-in implies wave-out_ what do they look like as a whole? What do these guarantees buy us if we _know_ the result will be a wave? For example, f(g(x)) is also guaranteed to be _wave-in-wave-out_. Again, maybe obvious, but it's a building block we can use once we've proved it true.
It appears so [1]. It's surprising how far behind most non-tech industries are when it comes to legacy software (not sure I'd call Apache "legacy", but still). I'd also be interested to see the stats for IT/developer positions in tech vs. non-tech fields. My intuition suggests that there are a lot more Apache sysadmins than you'd think.
Maybe there's insight there regardless? Hard to say. But the title could easily be "continuous functions are approximated by Fourier series with six terms" in my opinion.