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and construction of vector-valued wavelets. However, vector-valued wavelets [5] and
multiwavelets are different. Hence, studying vector-valued wavelets is useful in
multiwavelet theory and representations of signals. It is known that the majority of
information is multi-dimensional information. Thus, it is significant and necessary to
generalize the concept of multivariate wavelet packets to the case of multiple vector-
valued multivariate wavelets. Based on some ideas from [3], the goal of this paper is
to discuss the properties of multiple vector-valued multivariate wavelet packets.
Let R and C be all real and all complex numbers, respectively. Z and N denote,
respectively, all integers and all
positive integers.
Set
as well
Z
+
=
{0}
U
Nas N
,
,
∈
2
a
≥
2
a
elements
as
By algebra theory, it is obviously follows that there are
2
dd d
−
,, ,
a
L
Z
=
{(
zz zz Z
,
) :
,
∈
}
2
2
in
such that
Z
=
U
(
daZ
+
)
;
01
2
+
12
12
+
1
d
∈Ω
0
2
2
(
d Z d Z
+
)
I
(
+
)
=
φ
, where
Ω=
{,, ,
dd d
−
L
}
denotes the aggregate of all
1
2
0
0
1
2
a
1
2
2
the different representative elements in the quotient group
Z
/(
aZ
)
and order
dd
denote two arbitrary distinct
d
=
{
0
}
where {
0
} is the null element of
2
Z
+
and
12
0
Ω
elements in
.Let
Ω=Ω −
{
0
}
and
ΩΩ
,
to be two index sets. Define,
0
0
0
s
LRC
, we denote the set of all vector-valued functions
2
(,
By
)
LRC y hy hy hy hy LR l
2
( ,
2
s
):{() ( ()), (), , ()) : ()
=
h
=
L
T
∈ =
2
( ),
2
1,2, ,}
L
s
, where
1
2
u
l
T
means the transpose of a vector. For any
2
2
s
h
∈
LRC
(,
)
its integration is defin-
∫
∫
∫
∫
ed as
h
()
ydy
=
(
h ydy h ydy
() ,
() , ,
L
h ydy
() )
, and the Fourier tra-
2
2
1
2
2
2
s
R
R
R
R
h
()
y
nsform of
is defined by
ˆ
():
∫
h
ω
=
h
()exp{
y
⋅
−
i y
〈 〉
, }
ω
dy
, (1)
2
R
〈 〉
denotes the inner product of
y
and
y
,
ω
ω
where
. For multiple vector-valued
<Λ>
h
,
functions
denotes their symbol inner product, i.e.,
h h
, (2)
where
∗
means the transpose and the complex conjugate.
*
〈
,
Λ
〉
:
=
R
yyy
() ()
Λ
2
2
2
s
Definition 1.
A sequenc
{()
h
y
∈
⊂
L R C
(,
}
is called an orthogonal set, if
n
2
nZ
2
〈 〉
hh
nv nv s
I nvZ
,
=
δ
,
,
∈
, (3)
,
I
stands for the
ss
×
δ
where
identity matrix and
, is generalized Kronecker
nv
δ =
1
as
nv
=
δ =
0
symbol, i.e.,
and
, otherwise.
nv
,
nv
,
h
Definition 2.
A sequence of vector-valued functions
{( }
y
∈
⊂⊂
W
LRC
2
2
(,
s
)
is
n
2
nZ
called a Riesz basis in
W
, if it satisfies
Λ∈
(
yW
, there exists a unique
ss
×
{
n
nZ
P
(i) For any
matrix sequence
2
∈
such that
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