n this section we present a procedure for construction of dual wavelets with
orthogonality across scales. The disadvantage of this construction is
noncompactness of support for one of the wavelets
.
In the section
(
Compactly
supported smooth biorthogonal wavelets section
) we present a pair
with compact support but without orthogonality across scales (property (d)
below).
Proof
of (b),(c),(d). We define
and
the same way we did in the definition (
Dual
wavelets
). We get biorthogonality of
and
by the proposition (
Dual wavelets
properties
)f. We also have (
Dual
wavelets
properties
)a,d,e:
and (a) of the present
proposition:
From
and
we get
and
Then, by the formula (
Property
of scale and transport
2
),
and by
Then, by the formula (
Property
of scale and transport
2
),
Thus (b) and (c). The (d) follows because, by proposition
(
Dual wavelets properties
)f,
and
are bases in
.
