B-splines are symmetrical, bell-shaped functions
constructed from the (
n+1
n
1
) fold convolution of a rectangular pulse
β
0
x
β
0
x
:
β
0
x={1 if |x|<0.50.5 if |x|=0.50 otherwise
β
0
x
1
x
0.5
0.5
x
0.5
0
β
n
x=
β
+
,
1
0
*
β
+
,
2
0
*…*
β
+
,
n
0
*
β
+
,
n
+
1
0
x
β
n
x
β
+
,
1
0
β
+
,
2
0
…
β
+
,
n
0
β
+
,
n
+
1
0
x
Polynomial splines with equally spaced knots can be
uniquely characterized in terms of a B-spline expansion.
That is, for a spline
sx
s
x
of degree nn, there
exist unique coefficients
ck
c
k
such that
sx=∑k=−∞∞ck
β
n
x−k=c*
β
n
x
s
x
k
c
k
β
n
x
k
c
β
n
x
Understanding B-splines can help us better understand splines,
since B-splines are the building blocks.
-
The derivative of a B-spline of order
nn with respect to
xx is equal to the sum of two
B-splines of order
n−1
n
1
evaluated at shifts of
xx
ddx
β
n
x=
β
n
-
1
x+0.5−
β
n
-
1
x−0.5
x
β
n
x
β
n
-
1
x
0.5
β
n
-
1
x
0.5
-
The integral of a B-spline of order
nn is equal to the infinite sum
of B-splines of order
n+1
n
1
.
∫−∞x
β
n
tdt=∑k=0∞
β
n
+
1
x−0.5−k
t
x
β
n
t
k
0
β
n
+
1
x
0.5
k
-
Notice that this Fourier transform is related to the
n+1
n
1
fold convolution construction of the B-splines.
β
^
n
ω=sinω2ω2n+1
β
^
n
ω
ω
2
ω
2
n
1
-
B-splines are compactly supported.
-
They are the shortest possible polynomial splines.
-
β
n
x
β
n
x
is a piecewise polynomial of degree
nn:
β
n
x=1n!∑k=0n+1n+1k−1kx−k+n+12n
β
n
x
1
n
k
0
n
1
n
1
k
1
k
x
k
n
1
2
n
where
x
+
n={xn if x≥00 if x<0
x
+
n
x
n
x
0
0
x
0
