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The multiresolution analysis method is an important approach to obtaining wave-
lets and wavelet packs. We introduce the notion of multiresolution analysis of
2 4
L(R )
.
∈
Let ( )
φ
x
2 4
L (R ) satisfy the following refinement equation:
∑
φ
()
x
=⋅
a
d u
()(
φ
x
−
u
)
(1)
4
uZ
∈
is a real number sequence and
()
φ
x
where
is called a scaling function.
Taking the Fourier transform for the both sides of refinement equation (1), we have
{()}
uZ
du
4
∈
φ
(
AD
ωωφ ω
)
=
()()
(2)
2
4
U
⊂
L(R )
(
lZ
∈
Define a subspace
) by
l
U l s
=
|det( )|
A
l
φ
(
AxnnZ
l
−
):
∈
4
, (3)
()
l
24
LR
ϒ
()
x
2 4
L(R ) , if
We say that
in (1) generates a multiresolution analysis
{
jjZ
of
U
∈
the sequence
{
llZ
U
UU
+
⊂
∀
lZ
∈
,defined in (3) satisfies the below: (a)
,
;
∈
l
l
1
∩
∪
is dense in
=
{0}
fx U
()
∈⇔
fAx U
+
()
l
∈
2 4
L (R ) ; (c)
(b)
;
,
U
U
l
1
l
l
lZ
∈
lZ
∈
∀
lZ
∈
|det( )|
l
A
2
U
.
4
; (d) the family
{
φ
(
l
A xn n Z
−
) :
∈ is a Riesz basis for
}
(
)
W
lZ
∈
Let
denote the orthogonal complementary subspace of
U
in
U
+
1
and order
A
=
2,
I
also assume that there exists a vector-valued function
W
, i.e,
())
T
Fx
() ( (), (), ,
=
f x f x
f x
(see [7]) forms a Riesz basis for
1
2
15
W l s
=
f
(2
l
⋅ −
u
) :
ν
=
1, 2,
⋅⋅ ⋅
,15;
uZ l
∈
4
,
∈
Z
(4)
l
24
ν
LR
()
fx
,
1
()
fx
,
2
()
⋅⋅⋅
fx
15
()
∈
WU
⊂
From (4), it is clear that
,
. Therefore, there
0
1
()
ι
4
{( }
bu
( , ,
ι =
⋅⋅⋅
,
uZ
∈
)
exist fifteen real sequences
such that
∑
(
)
fx
() 16
=⋅
b u
()
ι
() 2
φ
x u
−
,
ι
∈Γ∈
,
u Z
4
.
(5)
ι
4
uZ
∈
Definition 1.
We say that a pair of vector-valued functions
ΦΦ∈
(), ()
x
x
LRC
2
( ,
4
v
)
are biorthogonal, if their translations satisfy
[( , (
Φ⋅ Φ⋅−
n
]
=
δ
I
,
n
∈
Z
4
,
(6)
0,
nv
I
denotes the
vv
×
δ
where
is the Kronecker symbol.
Definition 2.
A sequence of vector-valued functions
indentity matrix and
0
,
n
{( }
Τ
x
⊂
U
⊂
L
2
( ,
R
4
C
v
)
is
n
4
nZ
∈
called a Riesz basis of
U
if itsatisfies
(i)
for any
Gx
()
∈
U
, there exists a unique
2
4
vv
×
vv
×
{}
P
∈
ℓ
(
Z
matrix sequence
such that
n
4
nZ
∈
∑
Gx
()
=
PT x
(),
x R
∈
4
,
(7)
nn
4
nZ
∈
1
v
∑∑
ℓ
2
()
Z
4
vv
×
=
{:
P
Z
4
→
C
uu
×
,
P
=
2
where
|
qn
( ) | )
2
<+∞
,
(ii)
there
ls
,
2
ls
,
=
1
4
nZ
∈
exist two constants
0
<≤ <+∞
CC
such that, for any matrix sequence
{}
M
,
n
1
2
3
nZ
∈
the following equality follows.i.e.,
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