# Why do we use trig functions in Fourier transforms, and not other periodic functions?

Why, when we perform Fourier transforms/decompositions, do we use sine/cosine waves (or more generally complex exponentials) and not other periodic functions? I understand that they form a complete basis set of functions (although I don’t understand rigourously why), but surely other period functions do too?

Is it purely because sine/cosine/complex exponentials are convenient to deal with, or is there a deeper reason? I get the relationship these have to circles and progressing around them at a constant rate, and how that is nice and conceptually pleasing, but does it have deeper significance?

The Fourier basis functions $e^{i \omega x}$ are eigenfunctions of the shift operator $S_h$ that maps a function $f(x)$ to the function $f(x - h)$:
for all $x \in \mathbb R$.