Suppose that we wish to use the Haar wavelets to represent a familiar
function, such as 
. To do so, we can use the
familiar Hilbert space method of forming the Fourier-Haar coefficients
(because the 
are normalized). By writing out the
inner product 
explicitly as an integral, we have that
which becomes
Thus finding a Fourier-Haar coefficient for f is as simple as finding the difference of f itself integrated over two finite, adjacent intervals! Not surprisingly, this closely parallels the process used in the proof of convergence in the mean square metric.
Carrying through the process for the f chosen above, we find that the coefficients will be determined by
which, when integrated, yields
Using the terms generated by this function, we can see how well it
approximates 
in terms of 
for
. The domain of the original function is taken to be
. The result is shown in Figure 3.
Figure 3:
The Haar approximation of 
for m > -4.
We have not discussed any smooth functions that can mother orthogonal families. Two of these are shown and briefly described in Figure 4.
Figure 4:
Some mother functions suitable for generating orthogonal families.