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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?

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    $\begingroup$ 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. $\endgroup$ – Neal Jul 2 '16 at 19:47
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    $\begingroup$ Wavelets let you break a function into pieces that are localized in space and frequency. $\endgroup$ – mathematician Jul 2 '16 at 19:49
  • $\begingroup$ @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. $\endgroup$ – Chill2Macht Jul 2 '16 at 19:50
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    $\begingroup$ 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). $\endgroup$ – littleO May 18 at 22:28
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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:

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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.

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  • $\begingroup$ 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. $\endgroup$ – Chill2Macht Aug 4 '16 at 23:12
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    $\begingroup$ On on hand, it is really a completely different thing from Fourier analysis in the sense that the STFT decomposes a function into frequency and time components but for wavelets, it doesn't really make sense to take about frequency per se but rather it decomposes into the closely related notion of scale and time components. But to your point, it is indeed still an orthogonal decomposition (though there are more general formulations in terms of frames and such) and an integral transform so from that perspective they are very similar but if you zoom in, there are many subtle differences. $\endgroup$ – Wavelet Aug 4 '16 at 23:29

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