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For any α and the given vector-valued orthogonal scaling function
fy
, let
(
)
∑
Ψ=Ψ = Ψ− Ω
()
y
()
y
P
()
λ
(
ay n
),
λ
.
λ∈Ω (15)
α
a
β λ
+
n
β
0
0
2
nZ
∈
Definition 5.
The family of vector-valued functions
Ψ ∈ ∈Ω is
called vector-valued wavelet packs with respect to the vector-valued scaling function
()
{
βλ
β
(
y
),
Z
2
,
λ
}
a
+
+
0
fy
where
Ψ
()
y
is given by (16).
a
βλ
+
∑
()
μ
Ψ=Ψ = Ψ−
()
x
()
x
Q
(
ax
k
),
μ∈Γ
, (16)
α
a
σ μ
+
k
σ
0
2
kZ
∈
2
σ
∈
Z
ασμμ
=+ ∈Γ
a
,
where
is the unique element such that
follows.
+
0
%
v
FxFx LRC
2
2
Lemma 1
[4]
.
Let
(), ()
∈
( ,
).
Then they are biorthogonal if and only if
ˆ
ˆ
(
%
∑
*
Fk Fk
γ
+
2)
πγ
(
+
2)
π
=
I
.
(17)
s
2
kZ
∈
s
xLRC
2
2
μ∈
Γ
Lemma 2
[6]
.
Assume that
μ
Ψ∈
()
( ,
),
an orthogonal vector-valued
Ψ
0
()
x
μν∈Ω
,
wavelets associated with orthogonal scaling functions
. Then, for
,
0
we have
∑
P
()
μ
((
γ π
+
2
) /
a
)
P
()
ν
((
γ π δ
+
2
) /
a
)
*
=
I
.
(18)
μν
,
s
ρ
∈Ω
0
2
{( ),
Ψ
x
α
∈
Z
}
Lemma 3
[6]
.
Suppose
are wavelet packets with respect to or-
α
+
3
α
∈
Z
thogonal vector-valued functions
Ψ
0
()
x
. Then, for
, we have
+
[
Ψ⋅Ψ⋅− =
( ),
(
k
)]
δ
I
,
k Z
∈
2
.
(19)
α
α
0
,
ks
2
{( ),
Ψ
x
α
∈
Z
}
Theorem 1
[8]
.
Assume that
are wavelet packets with respect to
α
+
2
β
∈
,
Zv
μ
∈Ω
orthogonal vector-valued functions
Ψ
0
()
x
. Then, for
, we have
+
0
2
〈
Ψ⋅ Ψ⋅−
(),
(
m
)
〉
=
δδ
I m Z
,
∈
.
(20)
αβμ
a
+
a
β
+
v
0
,
k
μν
,
s
2
Theorem 2.
If
is vector-valued wavelet wraps with respect to a pair
of biorthogonal vector scaling functions
{( ,
Gx Z
β
β
+
∈
}
3
ασ
,
∈
Z
Gx
, then for any
0
()
, we have
+
%
2
[( ,
GGk I kZ
α
⋅
( ]
⋅−=
α
δδ
,
∈
.
(21)
σ
,
,
kv
ασ
=
ασ
≠
ασ∈Γ
,
Proof.
When
, (21) follows by Lemma 3. as
and
, it follows
0
α
β
from Lemma 4 that (21) 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
ααρ σσμ
=+ =+
a
,
a
ρμ∈Γ
,
, where
.
1
1
1
1
11
0
ασ
=
ρμ
≠
Case 1.
If
, then
. (21) follows by virtue of (19), (20) as well as
1
1
1
1
Lemma 1 and Lemma 2, i.e.,
1
ˆ
%
ˆ
%
∫
[( ,
GGk
α
⋅
( ]
⋅−
=
R
G
()
γ
G
() exp{
γ
*
⋅
ik
⋅
γ γ
}
d
σ
a
αρ
+
a
σμ
+
2
(2
π
)
2
11
11
1
∫
=
π
δ
I
⋅
exp{
ik
⋅
γ γ
}
d O
=
.
ρμ
,
s
(2
π
)
2
2
[0,2
]
11
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