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In-Depth Information
the functions in
V
1
which are not in any
V
1
e
t
. These hierarchical difference spaces
allow us the definition of a multilevel subspace splitting, i.e., the definition of the
space
V
n
as a direct sum of subspaces,
V
n
:¼
M
M
Þ
¼
M
jj
1
n
n
l
1
...
n
W
l
1
;...;l
d
W
1
:
ð
7
:
16
Þ
ð
l
d
Here and in the following, let
denote the corresponding element-wise
relation, and let
jj
1
:¼
max
ll
t
d
l
t
and |l|
1
:
¼
∑
t¼
d
l
t
denote the discrete
L
∞
-
and the discrete
L
1
-norm of l, respectively. As it can be easily seen from (
7.11
)
and (
7.15
), the introduction of index sets I
1
,
j
t
¼
1
,
...
,
2
l
t
1
,
j
t
odd
,
t ¼
1
,
...
,
d
,
if
l
t
>
0
,
N
d
I
1
:¼
ð
j
1
; ...;
j
d
Þ ∈
,
j
t
¼
0
t ¼
1
,
...
,
d
if
l
t
¼
0
,
,
,
ð
7
:
17
Þ
leads to
n
o
:
W
l
¼
span
ϕ
1
,
j
, j
∈
I
1
ð
7
:
18
Þ
Therefore, the family of functions
n
o
n
ϕ
1, j
,
j
∈
I
1
ð
7
:
19
Þ
0
is just a hierarchical basis [Fab9, Ys86] of
V
n
that generalizes the one-dimensional
hierarchical basis of [Fab9] to the
d
-dimensional case by means of a tensor-product
approach. Note that the support of all basis functions
ϕ
1,j
(x)in(
7.18
) spanning
W
1
is mutually disjoint.
Now, any function
f
∈
V
n
can be splitted accordingly by
f ðÞ¼
X
1
n
f
1
ðÞ¼
X
f
1
ðÞ
,
f
1
∈
W
1
¼ ϕ
1
,
j
ðÞ
,
and
j
∈
I
1
α
1
,
j
:ϕ
1,
j
ðÞ
,
ð
7
:
20
Þ
where
α
1, j
∈
R
are the coefficient values of the hierarchical product basis
representation.
It is the hierarchical representation which now allows to consider the following
subspace
V
ðsÞ
n
of
V
n
which is obtained by replacing
j
l
j
∞
n
by
j
l
j
1
n
+
d
1
(now with
l
t
>
0) in (
7.16
):
M
V
ðÞ
n
:¼
W
1
:
ð
7
:
21
Þ
jjnþd
1
