To have a concrete example as we introduce the
abstract framework later, we describe Haar wavelets.
Here is a picture of a Haar scaling function 
(x) and its
corresponding Haar wavelet 
(x):
[FIGURE]
First consider the space V0
L of all piecewise
constant functions f(x)
with breakpoints at integer values of x.
Clearly, f(x) can be written as a superposition of
integer translates of the Haar scaling function:
Thus the set {
(x-j) : j
} forms an
orthonormal basis for V0. In general, let Vi denote
the set of all piecewise constant functions with breakpoints
at x-values which are integer multiples 2i.
For instance, the function in
figure
is in V2.
The functions
forms an orthogonal basis for Vi. It becomes orthonormal if we use
instead.
Now let us consider the relationship among these subspaces. Clearly, we have
Consider the orthogonal complement Wi of Vi in Vi+1,
By definition, a function g(x)
Wi iff
for all f(x)Vi.
It is not hard to see that the functions
forms a basis for Wi.
EXERCISE: what are the breakpoints and support functions
for i,j and i,j?
Next, we can repeatedly expand the above equation to yield
In fact, we can continue this expansion for negative indices to yield
We note two other properties. Intuitively, it is clear that a function f(x)
L can
be arbitrarily closely approximated by functions in Vi
provided i
. This can be expressed by
saying that the set
is dense in L. Second, note that
where 0 is only the identically zero function.
