Proposition
(QMF property 2) Let
is QMF and
defined by the formula (
Definition of g_k
)
then
(a)
,
(b)
,
(c)
,
(d)
,
(e)
,
(f)
,
(g)
.
Proof
We will be using results of the section
(
Fourier analysis in
Hilbert space
) and the properties of scale and transport from the section
(
Elementary
definitions of wavelet analysis
) without further notice.
We prove
(a):
We prove (b) with the same calculation as
(a):
We prove
(c):
hence
To prove (d) we
calculate
The cases (e) and (f) are similar to (d). (g) is a collection of previous
statements.
