I'm wondering what series expansions exist for functions other than Taylor/MacLaurin and Fourier. I'm even more interested if there exists something in between, which use polynomials like Taylor (eventually wrapped in absolute values) but can approximate functions that are $C^0$ but not $C^1$, like Fourier (or even discontinuous functions altogether).
All other series expansions (or even things like step-function approximation in measure theory) are interesting to me in any case.
You may be interested in the Wavelet transform and corresponding series expansions. Yves Meyer has a good book about Wavelets. There is, of course, lots of information online about them. You can construct a Hilbert basis for $L^2$ with Wavelets, so smoothness isn't required.
Wiener's chaos expansion is also interesting and useful in stochastic analysis.