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?

• I would suggest the opposite is true: Fourier analysis focuses on the orthonormal basis of $L^2[0,1]$ provided by complex exponentials, and is a very specific (and well-developed and useful) subfield of the study of orthonormal bases in $L^2[0,1]$, which also encompasses wavelets. – Neal Jul 2 '16 at 19:47
• Wavelets let you break a function into pieces that are localized in space and frequency. – mathematician Jul 2 '16 at 19:49
• @Neal That makes sense. I've also heard people refer to the study of orthonormal bases in $L^2[0,1]$ as "Fourier analysis", whence my confusion. – Chill2Macht Jul 2 '16 at 19:50
• Wavelets are used, for example, in image compression and in reconstructing MRI images, based on the observation that certain wavelet transforms of natural images tend to be sparse (or nearly sparse). "Conceptually this isn't really an achievement" It's useful and it's not obvious (even to someone who already knows about the Fourier transform). – littleO May 18 at 22:28  • But that is a practical benefit. What about conceptually? As far as I can tell, it's still just $L^2$ decomposition, or a different way to do the same thing which has practical benefits and disadvantages. – Chill2Macht Aug 4 '16 at 23:12