Information Technology Reference
In-Depth Information
∑
Λ=
()
y
P
(),
nn
h
y
y R
∈
2
. (4)
2
nZ
∈
0
CC
<≤<∞
(ii) There exist constants
such that, for any constant matrix se-
1
2
{
n
nZ
P
qucnce
, we have,
2
∈
∑
h
(5)
CP Py y CP
∈
{ }
≤
() ()
≤
{ }
1
n
†
n
n
2
n
†
2
nZ
2
{}
P
{
n
nZ
P
Where
denotes the norm of sequence
2
†
∈
hD
in space
(), ()
x
x
We consider any functions
LR
with the inner product:
2
2
()
,
$
∫
()
x
−⋅
ω
h
()
ω
=
h
xe dx
∫
Dh
,
=
R
xxx
D h
( )
( )
.
2
2
R
As usual,
$
()
h
ω
h
()
LR
x
∈
2
2
()
denotes the Fourier transform of any function
.
2 Vector-Valued Multiresolution Analysis
Λ
Definition 3.
Let
Ω
be a separable Hilbert space and
be an index set. We recall
{
}
that a sequence
ι
Γ∈Λ⊆Ω
is a frame for
Ω
if there exist two positive constants
A
,
A
such that
2
2
∑
2
∀∈Ω
ξ
,
A
ξ
≤
ξ
,
Γ ≤
A
ξ
, (6)
1
ι
2
ι
∈Λ
A
,
A
are called frame bounds. A sequence
2
where
, and
ξ
=
ξ ξ
,
h
is
an exact frame if it ceases to be a frame when any one of its elements is removed.
If
{:
h
ι
ι∈Λ ⊆Ω
}
AA
=
. A frame
{:
ι
ι∈Λ
}
is a tight one if we can choose
1
2
AA
==
1
, then it follows from (1) that
,
1
2
∑
∀∈Ω
ξ
=
ξ
,
Γ Γ
ι
ι
ι
∈Λ
A sequence {
ι
Γ∈Λ⊆Ω
:
}
is a Bessel sequence if (only) the upper inequality of
()
()
1
follows. If only for all
h
∈⊂Ω
U
1
holds, the se-
, the upper inequality of
quence
{:
ι
Γ∈Λ⊆Ω
}
is a Bessel sequence with respect to (w.r.t.)
U
. More-
over, we assume that
U
⊂Ω
D
∈
U
is a closed subspace, if for all
, there exist two
2
∑
2
A
,
A
>
0
2
A
D
≤
|
D
,
Γ ≤
|
A
D
real numbers
such that
, the se-
2
1
ι
2
ι∈Λ
quence
{:
is called an
U
− subspace frame. In this section, we intro-
duce the notion of vector-valued multiresolution analysis.
Γ∈Λ
ι
ι
}
⊆Ω
Definition 4.
A vector-valued multiresolution analysis of
LRC
is a nested se-
2
(,
s
)
quence of closed subspaces {
X
}
such that (i)
XX j Z
+
⊂
1
,
∈
;
(ii)
j
j Z
∈
j
j
I
U
LRC
, where
O
denotes
ss
×
jZ
XO
=
{}
and
jZ
X
is dense in
2
(,
s
)
zero
j
j
∈
∈
h
()
yX yX j Z
+
h
( )
,
∈⇔ ∈ ∀∈
matrix; (iii)
; (iv) there is a vector-valued
j
j
1
Search WWH ::
Custom Search