<s>
In	O
the	O
mathematical	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
a	O
Bernstein	B-Algorithm
polynomial	I-Algorithm
is	O
a	O
polynomial	O
that	O
is	O
a	O
linear	O
combination	O
of	O
Bernstein	O
basis	O
polynomials	O
.	O
</s>
<s>
A	O
numerically	B-Algorithm
stable	I-Algorithm
way	O
to	O
evaluate	O
polynomials	O
in	O
Bernstein	B-Algorithm
form	I-Algorithm
is	O
de	B-Algorithm
Casteljau	I-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Polynomials	O
in	O
Bernstein	B-Algorithm
form	I-Algorithm
were	O
first	O
used	O
by	O
Bernstein	O
in	O
a	O
constructive	O
proof	O
for	O
the	O
Weierstrass	O
approximation	O
theorem	O
.	O
</s>
<s>
With	O
the	O
advent	O
of	O
computer	O
graphics	O
,	O
Bernstein	B-Algorithm
polynomials	I-Algorithm
,	O
restricted	O
to	O
the	O
interval	O
[0,1],	O
became	O
important	O
in	O
the	O
form	O
of	O
Bézier	O
curves	O
.	O
</s>
<s>
is	O
called	O
a	O
Bernstein	B-Algorithm
polynomial	I-Algorithm
or	O
polynomial	O
in	O
Bernstein	B-Algorithm
form	I-Algorithm
of	O
degreen	O
.	O
</s>
<s>
The	O
derivative	B-Algorithm
can	O
be	O
written	O
as	O
a	O
combination	O
of	O
two	O
polynomials	O
of	O
lower	O
degree	O
:	O
</s>
<s>
The	O
k-th	O
derivative	B-Algorithm
at	O
0	O
:	O
</s>
<s>
The	O
k-th	O
derivative	B-Algorithm
at	O
1	O
:	O
</s>
<s>
A	O
Bernstein	B-Algorithm
polynomial	I-Algorithm
can	O
always	O
be	O
written	O
as	O
a	O
linear	O
combination	O
of	O
polynomials	O
of	O
higher	O
degree	O
:	O
</s>
<s>
uniformly	B-Algorithm
on	O
the	O
interval[0,1]	O
.	O
</s>
<s>
Bernstein	B-Algorithm
polynomials	I-Algorithm
thus	O
provide	O
one	O
way	O
to	O
prove	O
the	O
Weierstrass	O
approximation	O
theorem	O
that	O
every	O
real-valued	O
continuous	O
function	O
on	O
a	O
real	O
interval	O
 [ a , b ] 	O
can	O
be	O
uniformly	B-Algorithm
approximated	O
by	O
polynomial	O
functions	O
over	O
.	O
</s>
<s>
Moreover	O
,	O
this	O
relation	O
holds	O
uniformly	B-Algorithm
in	O
x	O
,	O
which	O
can	O
be	O
seen	O
from	O
its	O
proof	O
via	O
Chebyshev	O
's	O
inequality	O
,	O
taking	O
into	O
account	O
that	O
the	O
variance	O
of	O
K	O
,	O
equal	O
to	O
x(1x )	O
,	O
is	O
bounded	O
from	O
above	O
by	O
irrespective	O
of	O
x	O
.	O
</s>
<s>
uniformly	B-Algorithm
in	O
x	O
.	O
</s>
<s>
uniformly	B-Algorithm
in	O
x	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
polynomials	O
fn	O
tend	O
to	O
f	O
uniformly	B-Algorithm
.	O
</s>
<s>
Bernstein	B-Algorithm
polynomials	I-Algorithm
can	O
be	O
generalized	O
to	O
dimensions	O
–	O
the	O
resulting	O
polynomials	O
have	O
the	O
form	O
.	O
</s>
<s>
In	O
the	O
simplest	O
case	O
only	O
products	O
of	O
the	O
unit	O
interval	O
are	O
considered	O
;	O
but	O
,	O
using	O
affine	B-Algorithm
transformations	I-Algorithm
of	O
the	O
line	O
,	O
Bernstein	B-Algorithm
polynomials	I-Algorithm
can	O
also	O
be	O
defined	O
for	O
products	O
.	O
</s>
