Wavelets are a powerful statistical tool which can be used for a wide range of applications, namely
One of the main advantages of wavelets is that they offer a simultaneous localization in time and frequency domain. The second main advantage of wavelets is that, using fast wavelet transform, it is computationally very fast.
You can learn more about wavelets in the following overview. We advise you to consult e.g. Daubechies (1992), Kaiser (1995) or Härdle et al. (1998) if you wish to study wavelets in more detail.
Wavelet expansion of a certain function is a special case of an
orthonormal series expansion.
Orthonormal series expansions of functions, or more generally
transformations in the frequency domain, have several important
applications. First, this is simply an alternative way of describing a
signal. Such a description in the frequency domain often provides
a more parsimonious representation than the usual
one on a grid in the time domain. Orthonormal series expansions
can also serve as a basis for nonparametric estimators and much
more.
You can recall the classical Fourier series expansion of a
periodic function. A nonperiodic function in can be
described by the Fourier integral transform which is represented by
an appropriately weighted integral over the harmonic functions.
A first step towards a time-scale representation of a function with
a time-varying behavior in the frequency domain is given by the
windowed Fourier transform, which approximately amounts to a
local Fourier expansion of . To attain time localization, one
multiplies
with a time window
and performs the usual Fourier
integral transform. For the windowed Fourier transform,
In contrast, the wavelet transform provides a decomposition into components from different scales whose degree of localization is connected to the size of the scale window. This is achieved by translations and dilations of a single function, the so-called wavelet. It provides an amount of simultaneous localization in time and scale domain which is in accordance to the so-called uncertainty principle. This principle basically says that the best possible amount of localization in one domain is inversely proportional to the size of the localization window in the other domain. Note that we do not use the term ``frequency'' in connection with wavelet decompositions. The use of this term would be adequate in connection with functions that show an oscillating behavior. For wavelets it is more convenient to use the term ``scale'' to describe such phenomena.
A function in
can be represented by its continuous
wavelet transform
There exist wavelets with several additional interesting properties. A frequently used tool are wavelets with compact support, which were first developed by Daubechies (1988). Moreover, one can construct wavelets of any given degree of smoothness and with an arbitrary number of vanishing moments. Finally, there exist boundary corrected bases which are appropriate for a wavelet analysis on compact intervals. This turns out to be quite a technical matter and you can consult the literature Daubechies (1992), Kaiser (1995), Härdle et al. (1998), if you are interested in more details.
The principle of estimation of a function in the space by
wavelets, the underlying multiresolution analysis (MRA), and
the meaning of father and mother wavelets, will be explained
also in the context of function approximation in Section 14.3.