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Theorem 2.
If
{( ,
Gx
β
∈
Z
3
}
and
{( ,
Gx
β
∈
Z
3
}
are vector-valued wavelet wraps
β
+
β
+
Gx
and
0
()
Gx
0
()
with respect to a pair of biorthogonal vector scaling functions
,
ασ
,
∈
Z
3
then for any
, we have
+
[( ,
GG k
⋅
( ]
⋅−=
α
δδ
I
,
kZ
∈
3
.
(26)
α
σ
,
,
kv
ασ
=
ασ
≠
ασ∈Γ
,
Proof.
When
, (26) follows by Lemma 3. as
and
, it follows
0
α
β
from Lemma 4 that (26) holds, too. Assuming that
is not equal to
, as well as at
least one of
{, }
ασ
Γ
ασ
,
doesn't belong to
, we rewrite
as
0
ααρσσμ
=+ =+
4
,
4
ρ μ ∈Γ
,
, where
.
1
1
1
1
11
0
ασ
=
ρ
≠
μ
Case 1.
If
, then
. (26) follows by virtue of (24), (25) as well as
1
1
1
1
Lemma 1 and Lemma 2, i.e.,
1
ˆ
ˆ
∫
*
[( ,
GG k
α
⋅
( ]
⋅−
=
G
()
γ
G
() exp{
γ
⋅
ik
⋅
γ γ
}
d
σ
4
αρ
+
4
σμ
+
(2
)
3
π
3
R
11
11
1
∫
=
π
δ
I
⋅
exp{
ik
⋅
γ γ
}
d
=
O
.
ρμ
,
v
(2
π
)
3
3
[0,2
]
11
3
ασ
≠
ααρσσμ
=+ =+
4
,
4
,
ασ
,
∈
Z
Case 2.
If
, order
where
, and
1
1
1
2
2
1
2
2
22
+
ρ μ ∈Γ
,
.
ασ
=
ρ
≠
μ
2
.
If
, then
Similar to Case 1, (28) follows. As
22
0
2
2
2
ασ
≠
ααρσ
=+
4
,
=
3
4
σμ
+
,
3
, order
where
ασ
,
∈
Z
,
ρμ
,
∈Γ
.
2
2
2
3
3
2
3
33
+
33
0
κ
α ∈Γ
ρ μ ∈Γ
,
.
Thus, taking finite steps (denoted by
), we obtain
, and
0
κκ
0
ˆ
ˆ
3
∫
*
ik
⋅
γ
8 [
π
GG k
(),
⋅
(
⋅−=
)]
GG
()
γ
()
γ
⋅
ed
γ
α
σ
α
σ
3
R
1
ˆ
ˆ
∫
*
=
G
()
γ
G
() exp{
γ
⋅
ik
⋅
γ γ
}
d
=
4
αλ
+
4
βμ
+
3
R
11
1
1
κ
Q
∏
∏
κ
∫
()
ρ
l
()
μ
l
*
=
{
Q
(
γ
/ 4 )}
⋅
O
⋅
{
(
γ
/ 4 )}
⋅
exp{
− ⋅
ik
γ γ
}
d
=
O
.
l
l
κ
3
l
=
1
([0,2 4
⋅
π
]
l
=
1
3
ασ
,
∈
Z
Therefore, for any
, result (26) is established.
+
, the translation operator
S
is defined to be
3
For any
vvvv Z
=
(, , )
∈
123
(
)( )
(
)
, where
a
is a pasitive constant real number.
S
x
=−
x
va
va
and
()
φφ
(), (), ()
xx x
ι
,
x
ι ∈
J
()
Theorem 3
[7]
.
Let
LR
de-
fined by (30), (31), (33) and (34), respectively. Assume that conditions in Theorem 1
are satisfied. Then, for any function
be functions in
2
ι
fx
()
LR
∈
()
, and any integer n,
2
7
n
−
1
∑
∑ ∑ ∑
f
,
φφ
( )
x
=
f
,
. (27)
( )
x
ι
:,
su
nu
,
nu
,
ι
: ,
su
3
3
uZ
∈
ι
= ∞
1
s
uZ
∈
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