What are wavelets?   What are they used for?

Wavelets (wavelet expansions and wavelet transforms) are one of the more
important modern improvements of Fourier series.

We start with a very carefully and cleverly chosen function, sometimes
called the "mother wavelet". Then we use it to generate a "wavelet
sequence", i.e. a sequence of functions which are all simply
translations and dilations of the mother wavelet. This sequence turns
out to be an orthonormal basis of L^2, with many interesting properties.

We all know how important and useful Fourier series and Fourier
transforms are in both pure and applied mathematics. But they have their
limitations. Here are some examples of where wavelet expansions can do better:

(1) In applied mathematics: If you try to approximate a function of one
or two variables, (i.e. an audio signal or a picture respectively) by
partial sums of the Fourier series of the function) this works nicely if
the function is fairly smooth, and gives a good compressed "imitation"
of the signal or picture which can be sent relatively easily by
international telephone line or stored in a relatively small computer
file. But if the signal has various noisy patches or the picture has a
number of rough spots, the Fourier method is much less successful in
compressing the data. In such situations wavelets do a much better job,
for example they are used to generate the newer kind of JPEG image
files, and for recent computer generated animated movies.

(2) In pure mathematics: In many situations in analysis it is very
helpful to be able to deduce properties of a function from properties of
its Fourier transform or Fourier coefficients, and vice versa.

But, there are very few natural properties of functions, (the exceptions
are being square integrable, or belonging to the special
Sobolev spaces W^{k,2}) which are exactly equivalent to some natural
property of the Fourier coefficients or transform of the function.

In contrast to this, many natural properties of functions ARE equivalent
to natural properties of the coefficients of the wavelet expansions of
these functions. Suitable sequences of wavelets can simultaneously act
as a natural basis for many different function spaces.