(Dual wavelets) Let
and
be dual GMRA (see the definition (
Dual GMRA
)) and
are the corresponding sequences defined in the proposition
(
Scaling equation
). We define the "wavelet"
and "dual wavelet"
according to the
formulas

Proposition

(Scaling equation 5) In context of the
definition (
Dual wavelets
) we
have
where

(c) The method of the proof if to verify condition of the proposition
(
Frame property 2
)-1. We
calculate
using the proposition (
Scaling equation
5
):
We aim to use 1-periodicity of
,
hence we separate even and odd
terms:
The
comes from a GMRA, hence
is a Riesz basis and (
Frame property 2
)-2
applies to each
:
Hence, we need an estimate for
.
The
already has the frame property by (
Frame property
2
)-2:
We repeat the same calculation for
starting from the left
inequality:
thus
Similarly, starting from
we
get
Combining these two inequalities with
completes the proof of (c).

It remains to verify that
and
are biorthogonal. First, we verify for the same scale index
:
We use the formula (
Property of
scale and transport
2
).
We use
(b).

We now verify orthogonality across scales. Let
.
We
have
By the formula (
Property of
scale and transport 2
) and (d) we
have
Hence
for any
and any
.