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1.4.6 Wavelets
Starting with a coarse (global) approximation, the idea behind wavelets
is to successively refine the representation and store the difference
from one scale to the next

(1.4.6#eq.1) 
This is best illustrated with Haar wavelets in figure (1.4.6#fig.1),
showing that the piecewise constant approximation at a level V4
can be brought to the higher level V5 by adding a correction
W4.
Figure:
Successive approximation of
using Haar wavelets.

Appropriate for integral equations and best suited for the understanding,
Haar wavelets are however not practical for the evaluation of
derivatives in PDEs. In the spirit of the FFT, Mrs Daubechies proposed
a fast
linear discrete wavelet transformation (DWT)
called DAUB4











(1.4.6#eq.2) 
which is applied successively to the function
together with
the permutation until only the first two components remain. The inverse
is calculated by reversing the procedure and using the inverse matrix







(1.4.6#eq.3) 
Approximations with DAUB4 have the peculiar property that the
derivative exists almost everywhere. Even if wavelets are not smooth,
they still can represent exactly piecewise polynomial functions of
arbitrary slope  the cusps in the wavelets cancel out exactly!
Figure (1.4.6#fig.2) illustrates how the solution converges when the
first
DWT components of DAUB4
are taken to
a higher level of refinement with
components.
Figure:
Successive approximation of
using DAUB4 wavelets.

Apart from Haar and Daubechies wavelets, a whole family can be generated
with a cascading process that is nicely illustrated in the
Wavelet Cascade Applet.
Wavelets are still relatively new and remain a matter of active research
[29]: they have been proven useful in approximation theory, but
are still being developed both for differential and integral equations.
Clearly apparent in the figures (1.4.6#fig.1) and (1.4.6#fig.2) is
the difficulty of incorporating the boundary conditions. Another issue that
will become clear in the coming sections, is the difficulty of calculating
overlap integrals.
SYLLABUS Previous: 1.4.5 Harmonic functions
Up: 1.4 Numerical discretization
Next: 1.4.7 Sampling with quasiparticles
