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In-Depth Information
1.3 Bernstein Polynomial
Bernstein polynomial approximation of degree p to an arbitrary real valued
function
f
(
t
)is
p
f
(
i
B
p
[
f
(
t
)] =
p
)
φ
ip
(
t
)
0
≤
t
≤
1
,
(1.1)
i
=0
where the function
φ
is the Bernstein basis function. The ith basis function is
precisely given by
φ
ip
(
t
)=
p
i
t
i
(1
t
)
p−i
,
−
i
∈
[0
,p
]
.
(1.2)
Some of the elementary properties of
φ
ip
(
t
)are:
p
∀
i
∈
[0
,p
]:
φ
ip
≥
∀
t
∈
[0
,
1] :
φ
ip
(
t
)=1
.
(1)
0;
i
=0
1]:
φ
0
p
(0) = 1;
φ
(
i
)
p
!
(
p−i
)!
.
(2)
∀
i
∈
[1
,p
−
ip
(0) =
1] :
φ
(
r
)
1] :
φ
(
s
)
∀
r
∈
[0
,i
−
ip
(0) = 0;
∀
s
∈
[0
,p
−
i
−
ip
(1) = 0.
1]:
φ
(
r
)
(3)
∀
r
∈
[0
,p
−
pp
=0;
φ
pp
(1) = 1.
(4)
φ
(
p−i
)
ip
p
!
1)
p−i
(1) = (
−
i
)!
.
(
p
−
(5)
φ
ip
(
p
)=
i
i
i
(
p
i
)
(
p−i
)
>φ
ip
(
t
)if
t
i
p
.
Properties (2) and (3) imply that the end point values,
f
(0) and
f
(1),
are the only values that are interpolated by the Bernstein polynomial. From
the condition for
φ
ip
(
t
) listed above, the end-point derivatives of
B
p
can be
obtained as follows:
−
=
1)
r−i
r
i
f
(
i
r
d
r
dt
r
B
ip
[
f
(
t
)]
p
!
|
t
=0
=
−
(
p
)
(1.3)
(
p
−
r
)!
i
=0
and,
1)
i
r
i
f
(
p
r
d
r
dt
r
B
ip
[
f
(
x
)]
p
!
−
i
|
t
=1
−
)
.
(
(1.4)
(
p
−
r
)!
p
i
=0
Hence, the rth derivative at the end points,
t
=0and
t
= 1, is determined by
the values of
f
(
t
) at the respective end point and at the r points nearest to
that end point. Specifically, the first derivatives are equal to the slope of the
straight line joining the end point and the adjacent interior point.
Bernstein polynomials satisfy the Weierstrass approximation theorem, i.e.,
they converge uniformly, with increasing p, to the function they approximate.
Also,
B
p
(
f
(
t
)) is smoother than
f
itself if smoothness is measured in terms of
the number of oscillations about a given straight line. Despite all these inter-
esting features, Bernstein polynomials are never widely used to approximate
the minimal norm. This is because they converge very slowly to the uniform
norm.
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