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fy X
()
∈
function
such that
{(): (
n
fy fynnZ
=−∈
):
2
}
form a Riesz basis for
0
fy X
()
∈
fy X X
()
∈⊂
subspace
.Since
, by definition and (4) there exist a
0
0
1
finite supported constant
ss
×
{
n
nZ
M
matrix sequence
such that
2
∈
∑
fy Mfayn
∈
()
=
(
−
)
, (7)
n
2
nZ
ˆ
ˆ
fa
( )
ωωωω
=Μ
()( ,
f
∈
R
2
,
(8)
1
∑
−
in
〈 〉
,
ω
where
Μ=
()
ω
Me
⋅
. Equation (7) is called a refinement equation and
2
a
n
2
nZ
∈
j
UjZ
,
∈
fy
a vector-valued scaling function. Let
()
be the direct complementary
X
X
+
a
2
−
1
subspace of
in
and there exist
vector-valued func-
j
1
Φ
()
y
tion
Φ∈
()
yLRC
2
( ,
2
s
),
ρ
Ω, such that the translates and dilations of
form
ρ
ρ
U
, i.e.
a Riesz basis of
U ls annZ
ρ
=
〈
Φ⋅−∈ ∈
(
j
) :
2
,
ρ
〉
. (9)
j
22
s
LRC
(,
)
Φ∈⊂ Ω
()
yUV
,
ρ
a
−
2
1
Since
there exist
finite supported constant matrix
ρ
0
1
{
n
nZ
B
ρ
sequences
such that
2
∈
∑
Φ=
()
y
B f ay n
ρ
(
− Ω
),
ρ
, (10)
ρ
n
2
nZ
∈
fx LRC
()
∈
2
( ,
2
s
)
The vector-valued functions
is said to be an orthogonal one, if
2
〈
f
(),
⋅ ⋅−
f n I nZ
(
)
〉
=
δ
,
∈
, (11)
0,
ns
2
2
s
Φ∈
()
yLRC
( ,
),
ρ
Ω
We say that
are orthogonal vector-valued wavelets
ρ
Φ
()
y
associated with an orthogonal vector-valued scaling functions
if
ρ
2
{(
Φ− ∈ ∈Ω
ynnZ
,
,
ρ
}
U
, and
is a Riesz basis of
ρ
〈
f
(),
⋅Φ⋅−
(
n
)
〉
=∈Ω∈
0,
ρ
,
n Z
2
. (12)
ρ
〈
Φ⋅Φ⋅−
(),
(
n
)
〉
=
ρ
δδ ρμ
, ,
ns
I
∈Ω
. (13)
ρ
ρ
,
,
3 The Orthogonality Traits of Vector-Valued Wavelet Wraps
To introduce the vector-valued wavelet packets, we set
0
()
(
0
)
(
ρ
)
(
ρ
)
2
Ψ= Ψ=Φ
yf y y y
(),
()
()
PMPB nZ
=
,
=
,
ρ
∈Ω ∈
,
.
,
ρ
ρ
n
n
n
n
Then, the equations (7) and (10), which
fy
and
(
)
Φ
(
y
)
satisfy can be written as
ρ
∑
()
ρ
Ψ=
()
y
P
Ψ− Ω
(
ay n
),
ρ
.
(14)
ρ
n
0
0
2
nZ
∈
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