Recall from spline theory that
fractional Bsplines reproduce polynomials
of order less than or equal to the ceiling of the spline order. This means that any
polynomial of a certain order can be expressed as a linear combination of Bsplines; that is,
the Bsplines form a basis for polynomials. Specifically, for some
pp which is a nonnegative integer no greater than
α+1α1, we have
∑kk∈Zkp
β
+
α
x−k=xp+
a
p
,
1
xp−1+…+
a
p
,
p

1
x+
a
p
,
p
k
k
k
p
β
+
α
x
k
x
p
a
p
,
1
x
p
1
…
a
p
,
p

1
x
a
p
,
p
(1) The collection of all these polynomials for all
pp no greater than
α+1α1 forms a
basis for polynomials of order no greater than
α+1α1.
This combined with the UnserBlu Scaling Function/Spline
Factorization Theorem leads to a straighforward proof of the fact that scaling
functions reproduce polynomials up to a degree proportional to their smoothness.
Let
φ
0
φ
0
be any
distribution
such that
∫
φ
0
xdx=1
x
φ
0
x
1
adn
∫xi
φ
0
xdx<∞
x
x
i
φ
0
x
,
for all
i∈1…n
i
1
…
n
, where
n=⌈α⌉
n
α
. Then
φx=
β
+
α
*
φ
0
x
φ
x
β
+
α
φ
0
x
reproduces polynomials of degree lesser or equal to
n=⌈α⌉
n
α
.

First note that
∑kk∈Zkpφx−k=∑kk∈Zkp
β
+
γ

1
*
φ
0
(x−k)=
φ
0
*∑k=0p
a
p
,
k
xp−k
k
k
k
p
φ
x
k
k
k
k
p
β
+
γ

1
φ
0
x
k
φ
0
k
0
p
a
p
,
k
x
p
k
(2)
because Bsplines have been shown to reproduce polynomials. This can be further simplified by studying the convolution of
φ
0
φ
0
with a polynomial.

Now we have
φ
0
x*xp=∫−∞∞x−up
φ
0
ud u=∑k=0pcpkxp−k−1k∫−∞∞uk
φ
0
ud u=xp+
b
1
xp−1+…+
b
p
φ
0
x
x
p
u
x
up
φ
0
u
k
0
p
c
p
k
x
p
k
1
k
u
uk
φ
0
u
x
p
b
1
x
p
1
…
b
p
(3)
where bk=(cpk−1k∫−∞∞uk
φ
0
ud u)
bk
c
p
k
1
k
u
uk
φ
0
u