[carstimon]

Can't we just appreciate that both points of views are helpful? Instead of thinking that one is better, and saying that therefore, the other sucks.

Both of them explain different things. The "it's just a basis transformation" makes invertibility, additivity, etc. obvious. The spinning function things makes it clear why we want to do it and what it means to say, approximate by only the small frequencies. (The basis thing tells you nothing about why looking at just the low coefficients is helpful, sometimes. A basis doesn't care about ordering)

Except for orientability, I guess.