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X
=
(
span
{(
Ψ⋅−
M
j
n
) :
n
∈
Z
3
,
μ
∈Γ
),
jZ
∈
.
(8)
j
μ
Ψ∈⊂ Γ
()
xX Y
,
μ
Since
, there exist 63 finitely supported sequences of
μ
0
1
constant
vv
×
{}
n
B
μ
()
matrice
such that
4
nZ
∈
∑
Ψ=
()
x
B
()
μ
F
(
x
− Γ
n
),
μ
.
(9)
μ
n
3
nZ
∈
By implementing the Fourier transform for the both sides of (9) , we have
ˆ
ˆ
()
μ
3
Ψ=
(
M
γ
)
B
( )
γ
Φ∈
( ),
γ
γ
R
,
μ
∈Γ
.
(10)
μ
where
∑
⋅
B
()
μ
( )
γ
=
B
()
μ
⋅
exp{
− ⋅
in
γ
},
μ
∈Γ
.
m
(11)
n
3
nZ
∈
Fx Fx
(), ()
∈
L R C
23
( ,
v
)
If
are a pair of biorthogonal vector-valued scaling func-
tions, then
3
[( ,(
FF n
⋅
⋅−=
]
δ
I
,
nZ
∈
.
(12)
0
,
nv
are pairs of biorthogonal vector-
valued wavelets associated with a pairof biorthogonal vector-valued scaling functions
()
23
v
ΨΨ∈
(),
x
()
x
L
( ,
R
C
),
μ
Γ
We say that
μ
μ
Fx
and
Fx
()
3
, if the family
{(
Ψ− ∈ ∈Γ
xnnZ
),
,
μ
}
is a Riesz basis of subspace
μ
X
, and
3
[( ,
F
⋅Ψ ⋅− =
(
n
] 0,
μ
∈Γ ∈
,
n
Z
.
(13)
μ
3
[( ,
F
⋅Ψ ⋅− =
(
n
] 0,
μ
∈Γ ∈
,
n
Z
.
(14)
μ
()
μ
j
3
X
=
Span
{(
Ψ⋅−
M
n
) :
n
∈ Γ ∈
Z
, ,
μ
,
j
Z
.
(15)
j
μ
Similar to (5) and (9), there exist 64 finitely supported sequences of
vv
×
complex
constant matrice
{}
Ω
{},
k
B
μ
()
μ ∈Γ
Fx
()
Ψ
()
x
and
such that
and
sat-
k
μ
3
3
kZ
∈
kZ
∈
isfy the refinement equations:
∑
Fx
()
∈
=Ω
FMx k
(
−
),
(16)
k
3
kZ
3 The Biorthogonality Features of a Sort of Wavelet Wraps
Denoting by
Gx
()
=Ψ
(),
x Q
(
0
)
=Ω
,
Gx
()
=
FxG x
(),
()
= Ψ
(),
xGx
()
=
Fx
(),
μ
μ
k
k
0
μ
μ
0
QB
()
μ
=
()
μ
,
μ
∈Γ ∈
,
k
ZMI
3
,
=
4
α
∈
Z
3
Q
(
0
)
=Ω
,
. For any
and the
QB
()
μ
=
()
,
μ
k
k
k
v
+
k
k
Gx
and
0
()
Gx
0
()
given vector-valued biorthogonal scaling functions
, iteratitively
define, respectively,
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