Orthogonality with respect to the index
follows trivially from (b). To show orthogonality with respect to the index
we apply the operation
to (a),(c) and use the definition
(
Multiresolution analysis
)-4 to
conclude
Thus for any pair
,
,
we
have
hence

We use (d) and the definition
(
Multiresolution analysis
)-2 to prove
that
is a basis. Indeed, we
have
and

Proposition

(Existence of orthonormal
wavelet bases 2) Let
be an MRA (see the definition
(
Multiresolution analysis
)) with the
scaling function
and
is the sequence defined in the proposition
(
Scaling equation
).
Set
Then
is an orthonormal wavelet basis.

Finally, we verify
(
Existence of orthonormal
wavelet bases 1
)-(d) using the proposition (
OST
property 2
). We need to show that
we
have
According to the proposition (
OST property 2
)
it suffices to show
that
where
and
We put all together and obtain the
requirement
where we introduced the convenience
notations
We simplify the
equation:
and introduce the
functions
Thus
The requirement
may be restated as
Therefore, we restate the target of the proof as finding a function
such
that
We rewrite the above relationships in matrix
form:
and substitute the definition of
from the proposition (
Scaling equation
2
):
We introduce convenience notations
and
.
We
have
and we utilize the proposition (
Scaling
equation 3
),
,
We multiply
by
and obtain an equivalent
equation
This proves existence of needed
.