Wavelet transform

The wavelet transform is a transformation to basis functions that are localized in scale and in time as well (where the Fourier transform is only localized in frequency, never giving any information about where in space or time the frequency happens). The frequency (similar in that sense to Fourier-related transforms) is derived from the scale.
As basis functions one uses wavelets. These functions are scaled and convolved with the function you are analysing all over the time axis.
Regarding the discrete version of the wavelet transform, the big advantage over the Fourier transform is the temporal (or spatial) locality of the base functions (see also short-time Fourier transform) and the smaller complexity (O(N) instead of O(N log N) for the fast Fourier transform (where N is the data size)).

In the likeness of the uncertainty principle the restriction for wavelet transform resolution can be written down:
:
and this result better in times as compared to the Fourier transform

Important applications are:

image compression and video compression: wavelet compression

Solving differential equations

signal processing

Types of wavelet transforms:

continuous wavelet transform (CWT)

discrete wavelet transform (DWT)

fast wavelet transform (FWT)

wavelet packets

complex wavelet transform