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DILATION EQUATION
The scaling property

and the fact that {
(t-j): j
} is an orthogonal
basis for V0 implies that

is an orthogonal basis for Vi.
EXERCISE: show this.
In general, for any function g(x) and j,k
,
we define

Then the above shows that {
j,k: k
} is
an orthogonal basis for Vj.
Restricting attention to V1, this means

for some ck's. This is called the dilation equation.
We can of course obtain ck as
the projection of
onto sqrt(2)
(2x-k),

Let us now derive several properties of the ck's.
For each m
,

In proof, use the fact that 
(x)
(x-m)dt=
(m),
and substitute the dilation equation for
(x) and for
(x-m).

Proof sketch: use the fact that
and substitute the dilation equation for
(x).
Dilation in the Fourier domain is expressed by

where

where we write hk for 2-1/2ck.
In proof, we just take the Fourier transform of both
sides of the dilation equation to give us

The transformation of this infinite sum to the
above formula ()
actually require justification (see [3], p.54).
Iterating this, we obtain

Note that

by the normalization assumption

We may also assume
. Hence

This infinite product is defined if H(0)=1.
Even if H(0)
1,
the infinite product may be well defined (because of
fortuitious cancellation:
see [Collela] in [1])
1. H(omega) is a trigonometric polynomial if the ck's
are zero for |k|>N (some N).
2. The case where ck=0 for |k|>N can be shown to lead
to a unique compact solution
(t) to the dilation
equation. [Show]
3. The dilation equation can be viewed as a fixed point operator,
and it can be used to compute
(t). It is easy
to see from this iteration that the solution
(t)
is supported in the interval [N,-N].
4. It is instructive to illustrate all the above equations
with the case of Haar functions where c0=c1=(1)/(sqrt(2)).
Chee Yap
