14.3 Function Approximation
Imagine a sequence of spaces
This sequence
is called multiresolution
analysis of
if
and
is dense in
(R).
Further, let us have
such that for
we have
where the circle around plus sign denotes direct sum.
Then we can decompose the space
in the following way:
and we call
father and
mother wavelets.
This means that any
can be represented as a series
According to a given multiresolution analysis, we can approximate
a function with arbitrary accuracy. Under smoothness conditions
on
, we can derive upper bounds for the approximation error.
Smoothness classes which are particularly well suited to the study of
approximation properties of wavelet bases are given by the scale
of Besov spaces
. Here
is the degree of smoothness while
and
characterize the norm in which smoothness is measured.
These classes contain traditional Hölder and
-Sobolev smoothness classes, by setting
and
, respectively.
For a given Besov class
there exists the following
upper bound for the approximation error measured in
:
The decay of this quantity as
provides a
characterization of the quality of approximation of a certain
functional class by a given wavelet basis. A fast
decay is favorable for the purposes of data
compression and statistical estimation.
The following display provides an impression of how a discontinuous
function is represented in the domain of coefficients.
The display contains two plots: the upper shows a jump function,
the lower the mother wavelet coefficients corresponding to their
position in scale and time. The large coefficients are caused by the
discontinuity (center) and by boundary effects since we use a
periodic wavelet transform for a nonperiodic function.
You can examine the effects of approximating
a wide range of functions using various wavelet bases with the
following interactive menu.
- Change basis -- changes the type of the
wavelet decomposition. Possible choices are
Haar, Daubechies 4, Coiflet 2
- Change level -- changes the number of the
father wavelet coefficients on the first
resolution level. This number must be a power
of 2 in the range
- Change function -- allows us to examine other
functions
- Change view -- provides other views to the
mother wavelet coefficients
- Print -- prints the display