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transforms. Wavelet wraps, owing to their nice characteristics, have been wid-ely
applied to signal processing [6],code theory, image compression, solving integral
equation and so on. Coifman and Meyer firstly introduced the notion of univariate or -
thogonal wavelet wraps. Yang [7] constructed a-scale orthogonal multiwavelet wraps
that were more flexible in applications. It is known that the majority of information is
multi-dimensional information. Shen [8] introduced multivariate orthogonal wavelets
which may be used in a wider field. Thus, it is necessary to generalize the concept of
multivariate wavelet wraps to the case of multivariate vector-valued wavelets. The go
-al of this paper is to give the definition and the construction of bioorthogonal vector-
valued wavelet wraps and designt new Riesz bases of
23
(,
RC
.
v
)
L
2 The Preliminaries on Vector-Valued Function Space
∪
,
s
nv
∈
N
3
We begin with some notations. Set
Z
+
=
{0}
N
,
and
snv
,,
≥
2,
Z
=
3
{(
zz z
,
,
) :
r
zZr
∈=
,
, , ,
Z
+
=
{{(
zz z
,
,
) :
:
r
zzr
+
∈=
,
1, 2 , 3} .
For any
123
123
3
X
,
XX
⊂
R
XX x
+=
{
+
x
2
:
xX
∈
1
,
, denoting by 4 4:
Xx
=
x
∈
X
,
1
2
1
2
1
1
xX
∈
2
}
XX x xx
−==−
1
{{
:
xXx X
∈∈
,
}
. There exist
64
elements
0
,
,
μ
2
1
2
1
1
2
1
2
2
4
Z
+
by finite group theory such that
μ
,
in
,
μ
Z
3
=
∪
(
μ
+
MZ
3
)
;
m
=
det
()
M
,
1
63
μ
∈Γ
0
(
+
MZ
3
)
∩
(
+
MZ
3
)
=
Ø
, where
Γ=
{,
μ
μ
1
,
,
μ
63
}
denotes the set of all
μ
μ
1
2
0
3
3
Z
/(
MZ
)
and
μ μ
,
denote
different representative elements in the quotient group
12
two arbitrary distinct elements in
Γ
0
,
M
is a
33
×
matrix Set
μ =
0
, where
0
is
0
3
Z
+
. Let
Γ=Γ −
{0}
and
ΓΓ
,
be two index sets.By
LRC
, we
2
(,
v
)
the null of
0
0
denote
the
aggregate
of
all
vector-
valued
functions
Hx
(),
i.e.,
v
, where
T
means the transpose of a vector. Video images and digital films are examples of
vector-valued functions where
hx
2
(), , ()) :
hx
T
hx
()
∈
LR
2
( ),
3
l
=
1,2, ,
v
LRC
2
(,
3
):{()(( ,
=
Hx
=
hx
1
v
l
∅
l
x
()
in the above
Hx
denotes the pixel on the
( )
l
th column at the point
x
. For
2
3
v
Hx
()
∈
L R C
( ,
),
H
denotes the norm of vector-
=
∑
∫
v
valued function
.In the below * means the
transpose and the complex conjugate, and its integration is defined to be
Hx
, i.e.,
2
1 / 2
()
H
:(
| ()|
x
dx
)
h
l
l
=
1
3
R
∫
∫
∫
∫
T
H xdx
()
=
(
() ,
xdx
() ,
x dx
,
() ).
xdx
h
h
h
1
2
v
3
3
3
3
R
R
R
R
Hx
is defined as
()
∫
The Fourier transform of
H
():
γ
=
H x
()
⋅
e
−⋅
ix
γ
dx
,
where
3
R
x
⋅
denotes the inner product of real vectors
x
and
FH
∈
L
2
(,
R
3
C
v
)
. For
,
γ
γ
their
symbol in ner product
is defined by
∫
*
[( , ( ]:
FH
⋅
⋅ =
FxHx x
() () ,
(1)
s
R
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