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space
S
as the set of all functions that can be expressed by
f
(
t
)=
k
a
k
φ
k
(
t
).
This is called the span of the basis set. In many cases, the signal spaces are
taken as the closure of the space, spanned by the basis set. This closure tells
us the space contains not only all signals that can be expressed by a linear
combination of the basis functions, but also the signals that are the limit of
these infinite expansions.
8.2.1 Cardinal B-Spline Basis and Riesz Basis
Since in wavelets we consider functions in
L
2
(
IR
) and our objective in this
chapter is to study spline wavelets, we consider cardinal splines that are both
in
S
n
and and
L
2
(
IR
), i.e., in
S
n
∩
L
2
(
IR
). We now suppose that
V
0
is its
closure. This means
V
0
is the smallest closed subspace of
L
2
(
IR
) that contains
S
n
∩
V
0
.
For simplicity we have considered cardinal splines with sequence of integer
knots
Z
. Now we consider the space
S
n
of cardinal spline functions with knot
sequences 2
−j
Z
,
j
L
2
(
IR
). Since
M
n
has compact support, one can visualize
B
⊂
Z
. Since a spline function with knot sequence 2
−j
1
Z
is
also a spline function with knot sequence 2
−j
2
Z
, whenever
j
1
<j
2
,wecan
write a nested sequence
∈
S
−
1
n
S
n
⊂
S
n
⊂···
···⊂
⊂
of cardinal spline spaces, with
S
n
=
S
n
.Ifwelet
V
j
to denote the
L
2
(
IR
)
closure of
S
n
∩
L
2
(
IR
), then the nested sequence
V
n
−
1
V
0
⊂
V
1
⊂···
···⊂
⊂
of closed spline subspaces of
L
2
(
IR
). Thus, the nested sequence of subspaces
satisfies
V
j
=
L
2
(
IR
)
,
j
∈
Z
(8.13)
V
j
=
{
}
,
0
j
∈
Z
where the overhead bar indicates the closure.
We now write the
n
th order cardinal B-spline basis through the convolution
of
N
n
(
x
)=(
N
n−
1
∗
N
1
)(
x
)
=
1
0
(8.14)
N
n−
1
(
x
−
t
)
dt.
m
≥
2
N
−
1 is the characteristic function of the interval [0
,
1). Setting
M
n
=
N
n
for
n
≥
2, we can tell
N
n
is an
n
th cardinal spline function in
V
0
⊂
S
n
.The
cardinal B-spline basis
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