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to Shen Z [5]. Since the majority of information is multidimensional information,
many researchers interest themselves in the investigation into multivariate wavelet
theory. The classical method for constructing multivariate wavelets is that separable
multivariate wavelets may be obtained by means of the tensor product of some
univariate wavelets. But, there exist a lot of obvious defects in this method, such as,
scarcity of designing freedom. Therefore, it is significant to investigate nonseparable
multivariate wavelet theory. Nowadays, since there is little literature on biorthogonal
wavelet packets, it is necessary to investigate biorthogonal wavelet packs.
In the following, we introduce some notations.
Z
and
Z
+
denote all integers and
all nonnegative integers, respectively.
R
denotes all real numbers.
R
denotes the 2-
dimentional
Euclidean
space.
2
()
LR
denotes the square integrable function space.
z
=
2
2
Let
xxxR
=
(, )
∈
,
ω
=
(, )
ωω ∈
R
,
i
kkkZz e
ω
=
(, )
∈
2
,
=
−
2
,
2
12
1
2
1
2
1
e
ω
−
i
2
2
. The inner product for any functions
()
φ
and
x
and
()( (), ()
x
φ
x
x
∈
L R
2
(
2
))
x
are defined, respectively, by
()
the Fourier transform of
∫
∫
φ
,
=
φ
( )
x
( )
x dx
,
(
ω
)
=
( )
x e
−⋅
ix
ω
dx
,
2
2
R
R
x
. Let
R and C be all real and all complex numbers, respectively. Z and N denote, respec-
tively, all integers and all
positive integers.
Set
()
where
ωωω
⋅=
x
x
+
x
and
x
denotes the complex conjugate of
()
11
2 2
a
≥
2
as well as
By
Z
+
=
{0}
∪
Nas N
,
,
∈
2
a
elements
dd d
−
,, ,
a
in
algebra theory, it is obviously follows that there are
01
2
1
2
Z
=
{(
zz zz Z
,
) :
,
∈
}
2
2
2
such that
Z
=
∪
(
daZ
+
)
;
(
d Z
+
)
∩
+
12
12
+
1
d
∈Ω
0
2
(
dZ
φ
+
)
=
, where
Ω=
{,, ,
dd d
−
}
denotes the aggregate of all the different
2
0
0
1
2
a
1
2
2
representative elements in the quotient group
Z
/(
aZ
)
and order
d
=
{
0
}
where {
0
}
0
dd
denote two arbitrary distinct elements in
Ω
is the null element of
2
Z
+
and
.Let
12
0
s
LRC
, we denote the
2
(,
Ω=Ω −
{
0
}
and
ΩΩ to be two index sets. Define, By
)
0
0
T
set of all vector-valued functions
LRC x
2
(,
s
):
=
{()
,
=
(( ,( ,,( ):
hxhx hx
1
2
u
, where
T
means the transpo--se of a vector. For any
l
hx LR l
()
∈
2
( ),
2
=
1,2, ,}
s
2
2
s
∈
LRC
(,
)
∫
∫
its integration is defined as follows
()
xdx
=
(
h xdx
() ,
2
2
1
R
R
∫
∫
() )
T
hxdx hxdx
2
() , ,
.
2
2
s
R
R
2
2
s
Definition 1.
A sequence
{()
y
⊂
L R C
(,
}
is called an orthogonal set, if
n
2
nZ
∈
2
〈
nv nv s
I nvZ
,
〉 =
δ
,
,
∈
, (1)
,
I
stands for the
ss
×
δ
where
identity matrix and
, is generalized Kronecker
nv
δ
=
1
as
nv
=
δ
=
0
and
, otherwise.
symbol, i.e.,
nv
,
nv
,
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