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∑
Gx G
()
=
()
x
=
QG xk
()
μ
(4
−
),
μ ∈Γ
, (18)
α
4
σ μ
+
k
σ
0
3
kZ
∈
∑
Gx G
()
=
()
x
=
QG xk
()
μ
(4
−
),
μ ∈Γ
. (19)
α
4
σ μ
+
k
σ
0
3
kZ
∈
3
σ
∈
Z
ασμμ
=+ ∈Γ
4
,
where
is the unique element such that
follows.
+
0
2
3
v
Lemma 1
[4]
.
Let
Fx Fx
(), ()
∈
L R C
( ,
).
Then they are biorthogonal if and only if
ˆ
ˆ
(
∑
F
γ
+
2)
k
πγ
F
(
+
2)
k
π
*
=
I
.
(20)
v
3
kZ
∈
Definition 4.
We say that two families of vector-valued functions
{
Gx
σμ
σ
+
( ),
∈
Z
3
,
4
+
μ ∈Γ
0
}
μ ∈Γ
0
}
and
{
Gx
( ),
σ
∈
Z
3
,
are vector-valued wavelet packets with
4
σμ
+
+
respect to a pair of biorthogonal vector-valued scaling functions
Gx
and
0
()
Gx
0
()
,
resp., where
Gx
σμ
()
and
Gx
σμ
()
are given by (18) and (19), respectively.
4
+
4
+
Applying the Fourier transform for the both sides of (18) and (19) yields, resp.,
ˆ
ˆ
G
( )
γ
=
Q
()
μ
(
γ
/ 4)
G
(
γ
/ 4),
μ
∈Γ
,
(21)
4
σμ
+
σ
0
ˆ
ˆ
G
(4 )
γ
=
Q
()
μ
( )
γ
G
( ),
γ
μ
∈Γ
,
(22)
4
σμ
+
σ
0
Gx Gx LRC
μ
(),
()
∈
23
( ,
v
),
μ ∈
Γ
Lemma 2
[6]
.
Assume that
are pairs of biort-
hogonal vector-valued wavelets associated with a pair of biorthogonal scaling func-
tions
μ
Gx
and
0
()
Gx
0
()
μ ν ∈Γ
,
. Then, for
, we have
0
∑
()
μ
()
ν
*
Q
((
γ
+
2
π
) / 4)
Q
((
γ
+
2
π
) / 4)
=
δ
I
.
(23)
μν
,
v
ρ
∈Γ
0
3
3
{( ,
Gx
α
∈
Z
}
and
{( ,
Gx
α
α ∈
Z
+
}
Lemma 3
[6]
.
Suppose
are wavelet packets
α
+
Gx
and
0
()
Gx
0
()
with respect to a pair of biorthogonal vector-valued functions
.
3
α
∈
Z
Then, for
, we have
+
3
[( ,
GG k
⋅
( ]
⋅−=
δ
I
,
kZ
∈
.
(24)
α
α
0
,
kv
3
3
Z
+
are vector-
valued wavelet packets with respect to a pair ofbiorthogonal vector-valued functions
0
()
{( ,
Gx
β
∈
Z
}
and
{( ,
Gx
β
β ∈
}
Theorem 1
[8]
.
Assume that
β
+
Gx
and
Gx
0
()
β
∈
,
Zv
3
μ
∈Γ
, respectively. Then, for
, we have
+
0
[
GG k
( ),
⋅
(
⋅−
)]
=
δδ
I
,
k
∈
Z
3
.
(25)
4
βμ
+
4
β
+
v
0
,
k
μν
,
v
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