How is the study of wavelets not just a special case of Fourier analysis? 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?
 A: Shannon wavelets have dual basis functions resembling the reconstruction functions for Fourier Transforms. If you apply a dyadic subdivision on both frequency bands what you will get is something very similar to the FFT. In this sense is rather the FFT which is a special case of DWT.
Here we can see how wavelet and scaling functions complement each other for one  frequency band:


The wavelet functions look like and are related to the sinc-functions which famously do appear in Fourier analysis as the "reconstruction" functions. This is also visible in Fourier transform of the wavelet functions, they are very close to "boxes" which is what the FT of sinc functions are.
A: Not to dredge up old posts but comparing wavelets and the Fourier transform is really apples and oranges. A much better comparison is that of the STFT and the various wavelet transforms. As @mathematician mentioned, the benefit of wavelets over STFT is that wavelet analysis allows for us to (somewhat) sidestep the Gabor limit via a more flexible tiling of the time-frequency plane. 
