Bernstein Polynomial and B´ezier-Bernstein Spline - Bezier and Splines in Image Processing and Machine Vision - page 6

p
k

t k (1 p

k

t k (1

k∈A |

<
k∈A

t ) p−k t ) p−k f ( k/p ) − f ( t ) | − −

p
k

t k (1

p

(1.7)

t ) p−k

< −

k =0

< ,

since the extended sum is unity.
Let us now estimate the second sum where k is in set B . Since f is
continuous and [0 , 1] is compact, there is an M t such that | f ( t ) |≤ M t .

M t = | f ( t ) | max , 0 ≤ t ≤ 1. So, we get | f ( k/p ) − f ( t ) |≤ 2 M t consider-

ing the worst case (when f ( k/p )= − f ( t ) or when f ( k/p )and f ( t )areof

opposite sign). Therefore,

p
k

t k (1 p

k

t k (1

k∈B |

2 M t
k∈B

t ) p−k t ) p−k . f ( k/p ) − f ( t ) | − ≤ −

pt ) 2
p 2 δ 2

( k −

t ) 2 δ 2 If k is in B , then ( k/p − ≥ or ≥ 1. Now one can prove the

identity

pt ) 2 p
k

t k (1

p

p
4 .

t ) p−k

( k − − ≤ (1.8)

k =0

Using equation (1.8), we can show that

p
k

t k (1 p

k

t k (1

p

px ) 2
p 2 δ 2

( k −

t ) p−k t ) p−k . − ≤ −

k∈B k =0

The second sum is, therefore,

p
k

t k (1

t ) p−k

B |

f ( k/p ) − f ( t ) | −

k ∈

pt ) 2 p
k

t k (1

p 2 δ 2
k

2 M t

t ) p−k

≤ ( k − −

∈ B

pt ) 2 p
k

t k (1 (1.9)

p

2 M t
p 2 δ 2

t ) p−k

≤ ( k − −

k =0

p
4

2 M t
p 2 δ 2

≤

M t

2 pδ 2 .

=

Considering equations (1.6), (1.7), and (1.9) ∀ t ∈ [0 , 1] we can write,

M t
2 pδ 2 .

| B kp ( t ) − f ( t ) |≤ +

M t

Therefore, | B kp ( t ) − f ( t ) | < 2 whenever

2 pδ 2 < . Thus, we get,