As far as I can tell, "wavelets" is just a neologism for certain "non-smooth" families of functions which constitute orthonormal bases/families for $L^2[0,1]$.
How is wavelet analysis anything new compared to the study of Fourier coefficients or Fourier series or the orthogonal decomposition of $L^2$ functions (i.e. in the most abstract possible function analytic sense, not in the sense of using specifically the orthonormal families of sines/cosines or complex exponentials)?
Wavelet transforms just seem like the Fourier transform using a different orthonormal family for $L^2$ besides the complex exponentials, but conceptually this isn't really an achievement. The complex exponentials are a convenient orthonormal family, but at the end of the day aren't they just an orthonormal family?