Everything up to this point has been a review of Hilbert space theory. Note that the main value of this theory is that operators may be analysed in terms of their associated eigenspaces, i.e. in terms of known basis functions.
The Haar basis is an orthogonal basis which has been known since 1910 [Haar]. It is also the earliest known example of a wavelet basis, and perhaps one of the simplest. For these reasons we will use it in outlining the basics of wavelet analysis.
In Hilbert space analysis, one finds the union of eigenspaces of a linear operator and determines the basis of that space. The operator itself is chosen to reflect some aspect of the problem domain. For example, one might choose a 2nd-order ODE with constant coefficients to describe some sort of simple harmonic oscillator. The basis of the eigenspace is readily shown to be made of sine and cosine functions with definite periods.
Wavelet analysis provides other means to generate a basis suited to
the problem domain.
For example, the Haar basis may be generated from a
so-called ``mother function'' in formulaic fashion. This strategy is
used in a general fashion to create wavelet families.
One prescription used to generate a wavelet family 
from a
suitable function 
is
For some very special choices of 
and 
the
constitute an orthonormal basis for 
.
In particular, for 
there do exist
such that the
form an orthonormal basis of 
. Indeed, all but the last
chapter of [Daubechies] is concerned with bases arising from the
choice 
. This is the generating function we will use.
The Haar basis can be generated by these means from the Haar function
This function is shown in Figure 1.
To establish that the Haar family 
does indeed
constitute an orthonormal basis we must show that
are orthogonal and normalized, and
can be approximated to arbitrary precision
by a partial expansion in the 
.
First we check orthogonality. The support of 
is readily
seen to be 
, which leads immediately to the
conclusion that 
when 
.
Stated another way, Haar wavelets of the same scale m do not overlap
unless they share the same index of translation n. This leads to
the integral
.
Thus we have 
.
Checking orthogonality in the case of differing scale is a bit harder.
Given 
and 
the supports will be
and 
. The situation is
illustrated in Figure 2, for Haar wavelets 
and
(four wavelets of scale m=0).
It is easy to check that if 
then the support of 
lies wholly within a region where 
is constant.
Thus the integrand cancels itself in the integration. By switching
the wavelets we can show the same thing for 
.
Figure 2:
The Haar wavelets 
(solid)
and 
(dashed). Note that the height of
such a wavelet goes as the inverse square root of its width.
Combining this last
with the previous result concerning wavelet scale, we finally have
that 
.
Thus the 
are an orthonormal family of compactly supported
wavelets.
Now we would like to show that it is possible to expand any
in a finite number of terms of 
so that
for arbitrarily small 
. In other words, we are looking for
the partial sum to converge in the mean square metric. The formal
proof of this is a delightful exercise in the notation of intervals
and is presented very clearly in the book by [Daubechies].
We summarize the proof briefly.
Because any function 
is Lebesgue-integrable, we know
that f can be arbitrarily well approximated by a function with
compact support which is piecewise constant on intervals
(for sufficiently large support and
j). We may choose two adjacent intervals that exactly
overlap the interval of some Haar wavelet 
and represent
their difference by a constant 
. This process is repeated
until the support of f is covered by wavelets of scale -j+1. Then
we go back to choose constants 
and cover the support of f
with wavelets of scale -j+2.
This process is repeated until the support of the wavelet engulfs the support of f, at which point successive wavelets begin to represent the (diminishing) total error of the partial sum. This can be extended indefinitely, until the given error threshold is reached. Thus f can be arbitrarily well approximated by a partial series expansion of Haar wavelets.
.