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(
)
Lemma 2.
Assuming that
()
x
is an semiorthogonal scaling function.
Bzz
is
,
12
the symbol of the sequence
{( )}
bk
defined in (3). Then we have
2
2
2
2
Π=
Bz z B z z
(, )
+−
( , )
+
Bz z
(,
− +−−
)
B z z
( ,
)
(17)
12
12
1
2
1
2
2
x
is an orthogonal bivariate function, then
()
∑
(
)
Proof.
If
h
ωπ
+
2
1
.
2
kZ
∈
Therefore, by Lemma 1 and formula (2), we obtain that
=
∑
1
|
Be
(
−
i
(
ωπ
2 )
+
k
,
e
−
i
(
ω π
2
+
k
)
)
⋅
h
((
ωω
,
) 2
+
(
k k
,
)
π
) |
2
1
1
2
2
12
12
2
kZ
∈
∑
∑
=
|
Bz z
(, )
h
(
ωπ
+
2 )|
k
2
+
|
B z z
( , )
−
⋅
h
(
ωπ
+
2
k
+
,0))|
π
2
12
2
12
2
kZ
∈
kZ
∈
∑
∑
+
|
Bz z
(,
−⋅
)
h
(
ωπ
+
2
k
+
(0, ))|
π
2
+ −−⋅
|
B z z
( ,
)
h
(
ωπ
+
2
k
+
, ))|
π
2
1
2
2
1
2
2
kZ
∈
kZ
∈
=
2
2
2
2
B zz Bzz Bz z Bz z
(, )
+−
( , )
+
(,
− +−−
)
( ,
)
12
12
1
2
1
2
This complete the proof of Lemma 2. Similarly, we can obtain Lemma 3 from (3),
(8), (13).
Lemma 3.
If
ψ
()
x
ν =
0,1, 2, 3
(
) are orthogonal wavelet functions associated with
ν
hx
. Then we have
()
∑
1
()
λ
j
+
1
j
()
λ
j
j
()
ν
j
j
+
B
((
−
1)
z
, (
−
1)
z
)
{
B
((
−
1)
z
, (
−
1)
z B
)
((
−
1)
z
, (
−
1)
z
)
1
2
j
=
0
1
2
1
2
)
}:
=Ξ
=
δ
,
λ ν
,
∈
{0,1,2,3}.
()
ν
j
+
1
j
⋅
B
((
−
1)
z
, ( 1)
−
z
(18)
λμ
,
λν
,
1
2
For an arbitrary positive integer
nZ
+
∈
, expand it by
=
∑
∞
j
−
1
n
ν
4,
ν
∈Δ =
{0,1, 2, 3}
. (19)
j
j
j
=
1
Lemma 4.
Let
nZ
+
∈
and n be expanded as (17). Then we have
−
i
ω
−
i
ω
∞
()
1
2
()
∏
ν
()
Λ=
ω
Be e
(
j
,
j
)
Λ
0 .
j
2
2
n
0
=
1
j
Lemma 4 can be inductively proved from formulas (14) and (18).
Theorem 1.
For
nZ
+
∈
kZ
∈
3
,
, we have
Λ⋅ Λ⋅−
(),
(
k
)
=
δ
.
(20)
n
n
0,
k
Proof.
Formula (20) follows from (10) as n=0. Assume formula (20) holds for
the case of
0
≤<
4
r
r
is a positive integer). Consider the case of
(
0
4
r
≤<
n
4
r
+
1
ν ∈Δ
. For
, by induction assumption and Lemma 1, Lemma 3 and
0
0
Lemma 4, we have
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