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cardinal B-spline functions. We shall, therefore, discuss cardinal splines first
in this chapter. Readers can consult Schoenberg and Chui [147, 37] for an
extensive study on cardinal splines and their uses.
In this chapter, we shall restrict ourselves to spline wavelets and their
properties but to understand them well, we shall also discuss the related es-
sentials.
8.2 Cardinal Splines
Cardinal splines are polynomial spline functions with equally spaced knots.
Because of the simple knot structure, these splines can be used easily with
computational advantages. One of the major advantages of cardinal splines
over others is that cardinal splines have essentially only one B-spline of a
given order. All others of the same order are (scaled) translates of this one.
Further simplicity and convenience can be achieved if we consider that knots
are integers. Let us assume
n
is an integer,
n
≥
0and
S
n
=
{
f
(
x
)
}
, Class of
C
n−
2
(
IR
)and
f
(
x
)
polynomial functions of order at most n, with
f
(
x
)
∈
π
n−
1
.
π
n
is the collection of all algebraic polynomials of degree not exceeding
n
and
f
(
x
)are
n
∈
2 times differentiable. Elements of
S
n
are called cardinal
spline functions of degree
n
. Therefore, if we restrict our attention to any
interval [
j, j
+ 1) where j is an integer, then the function in [
j, j
+1)is
−
f
∈
π
n−
1
,
j
∈
Z.
We can now connect two polynomial pieces of functions in adjacent intervals.
Consider two intervals [
j, j
+1) and [
j
−
1
,j
). Let the polynomials in these two
intervals be
p
(
l
)
n,j
and
p
(
l
)
n,j−
1
from the collection of
π
n−
1
and
j
=
−
N,
···
,N
−
1.
Considering the continuity of the two polynomials at the point
x
=
j
, one can
write
p
(
l
)
p
(
l
)
2
.
Now the order of the polynomials is
n
. Hence the degree of each of the poly-
nomials is
n
n,j
(
j
)
−
n,j−
1
(
j
)=0
,
l
=0
,
1
,
···
,n
−
2
,
n
≥
1)th differentiation each of them is a constant that
is different at the knot sequence
Z
, as we approach it from the right and left
sides of
j
. This means we can write this difference as
−
1. After (
n
−
c
j
=
p
(
n−
1)
p
(
n−
1)
n,j
(
j
+)
−
1
(
j
−
)
n,j
−
(8.1)
f
(
n−
1)
(
j
+
)
f
(
n−
1)
(
j
= lim
0
{
−
−
)
}
.
→
c
j
is the jump of
f
(
n−
1)
and can be used to link between the polynomial
pieces in two adjacent intervals.
c
j
(
n−
1)!
can be taken as the leading coecient
of the difference polynomial between the two adjacent intervals. Note that
other coecients are zero. Hence
c
j
j
)
n−
1
.
p
n,j
(
x
)=
p
n,j−
1
(
x
)+
1)!
(
x
−
(8.2)
(
n
−
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