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∑
CM
{}||
≤
MHx C M
( ||
≤
{},
(8)
1
n
n
n
2
n
*
*
3
nZ
∈
2
4
are pairs of biorthogonal vector-
valued wavelets associated with a pairof biorthogonal vector-valued scaling functions
()
We say that
fxf x LR
μ
(),
()
∈
( ),
μ
∈ Γ
μ
φ
x
and
Fx
()
, if the family
{(
Ψ− ∈
xnnZ
),
3
,
μ
∈Γ
}
is a Riesz basis of subspace
μ
X
, and
4
[(),
φ
⋅
f
(
⋅ −=
n
)] 0,
μ
Γ∈
,
n
Z
.
(9)
μ
4
[(), (
φ
⋅
f
⋅ −=
k
)] 0,
μ
Γ∈
,
k
Z
.
(10)
μ
()
μ
j
3
X
=
Span
{(
Ψ⋅ −
M
n
) :
n
∈
Z
, ,
μ
Γ
,
j
∈
Z
.
(11)
j
μ
Similar to (5) and (9), there exist 64 finitely supported sequences of
vv
×
complex
constant matrice
such that
()
and
()
()
{},
k
B
μ
{}
d
μ ∈Γ
φ
x
f
x
and
satisfy
k
4
3
μ
kZ
∈
kZ
∈
the refinement equations:
∑
φ
()
x
=
d
φ
(
Ax
−
k
),
(12)
k
4
kZ
∈
()
ι
∑
(
)
4
fx
() 16
=⋅
b k
()
φ
2
xk
−
,
ι
∈Γ∈
,
k Z
.
(13)
ι
4
kZ
∈
3 The Traits and Features of a Sort of Quarternary Wavelet
Wraps
We begin with the following notations:
()
()
()
4
ι ∈Γ
uZ
∈
0
ι
ι
Γ=
0
()
x
φ
(),
x
Γ=
()
xf
()
x
,
du du
()
=
(),
du bu
()
=
(),
,
ι
ι
{
Γ
( ) :
xn
=
0,
1, 2,
⋅⋅⋅ ν
,
=
0,1, 2,
⋅⋅⋅
,15}
Definition 1.
A set of functions
is
called a nonseparable four-dimensional
wavelet wraps with respect to the orthogonal
scaling function
16
n
+
φ
()
x
, where
∑
Γ
()
x
=
d
()
ι
()
n
Γ
(2
x
−
k
),
v
∈ Λ
=
{0,1,2, 15},
(14)
16
nv
+
4
v
0
kZ
∈
By implementing the Fourier transform for the both sides of (14), we have
(
)
(
)
Γ
()
ω
=
Dzzzz
()
v
, , ,
⋅ Γ
ω
2.
v
∈Λ
. (15)
16
nv
+
n
1234
0
(
)
∑
()
ν
()
ν
()
ν
kk
k
k
Dzzzz D
,,,
=
(/ )
ω
=
dkzzzz
()
(16)
1
2
3
4
1 2 3 4
1234
4
kZ
∈
2 4
() L(R).
Lemma 1
[7]
. Let
hx
∈
Then ( )
φ
x
is an orthogonal function if and only if
∑
|
h
(
ωπ
+
2 )| =1
2
. (17)
4
uZ
∈
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