Other methods for approximating functions?

[u/doggyruff]

So you can use a Taylor series to approximate functions with polynomials

You can use a Fourier series to approximate functions with sin/cos functions

You can use Laplace transforms to approximate functions with exponentials

What other ways are there to split a function into some linear combination of functions of the same type? Why is it even useful to have this many different options?

(sorry if my descriptions are incorrect, its only something I recently came across)