Graphics Reference
In-Depth Information
N
1
(
x
)=1
,
0
≤
x<
1
,
(8.10)
=0
,
otherwise.
Now,
2
f
)(
x
)=(
(
(
f
))(
x
)
,
=(
(
f
(
x
)
−
f
(
x
−
1))
,
=(
f
)(
x
)
−
(
f
)(
x
−
1)
,
=(
f
(
x
)
−
f
(
x
−
1))
−
(
f
(
x
−
1)
−
f
(
x
−
2))
,
1)
k
2
k
(
x
2
=
(
−
−
k
)
+
,
k
=0
where,
f
(
x
)=(
x
−
0)
+
−
1,
f
(
x
−
1) = (
x
−
1)
+
−
1and
f
(
x
−
2) = (
x
−
2)
+
−
1
for
n
= 2. Proceeding this way, one can easily show that
1)
r
n
r
(
x
n
1
r
)
n−
1
+
M
n
(
x
)=
(
−
−
.
(8.11)
(
n
−
1)!
r
=0
Obviously,
M
n
(
x
)=0for
x
n
. Also,
M
n
(
x
)=0for
x<
0 (since,
x
+
=
max
(0
,x
)). This helps to establish
≥
supp M
n
=[0
,n
]
.
With this, we observe that:
(1) The collection
B
=
{
M
n
(
x
−
r
)
,r
∈
Z
}
reduces to
B
2
=
{
M
n
(
x
−
r
)
,r
=
−
N
−
n
+1
,
···
,N
−
}
1
.
(2)
M
n
(
x
−
r
)=0for
r>N
−
1and
r<
−
N
−
m
+1.
(3)
is a linearly independent set.
Hence,
B
2
is a basis of
S
n,N
. We can take the union of
S
n,N
over
N
=1
,
2
,
3
,
{
M
n
(
x
−
r
)
}
···
and we come to
B
. This helps to write a spline series as
∞
f
(
x
)=
a
r
M
n
(
x
−
r
)
.
(8.12)
r
=
−∞
We shall now describe the importance of the space
L
2
(
IR
) and the basis set
from the engineering point of view.
L
2
(
IR
) space is important in signal pro-
cessing. This is the space of all functions
f
(
t
), which can be used to represent
a signal. The energy of the signal can be taken as the integral of the square
of the modulus of the function. Since, this integral is finite, it corroborates
the fact of finite energy of a signal in practice.
IR
indicates the time instant
t
of occurrence of the signal (also the independent variable of integration) is a
number on the whole real line.
Now if we start with the vector space of signals
S
, then if any
f
(
t
)
∈
S
can
be expressed as
f
(
t
)=
k
a
k
φ
k
(
t
), then the set of functions
φ
k
(
t
) is called an
expansion set for the space
S
. If the representation is unique, then the set is a
basis. One could also start with the expansion set or basis set and define the
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