BERNSTEIN POLYNOMIAL

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:''For the Bernstein polynomial in D-module theory, see Bernstein-Sato polynomial.''

In the mathematical subfield of numerical analysis, a 'Bernstein polynomial', named after Sergei Natanovich Bernstein, is a polynomial in the 'Bernstein form', that is a linear combination of 'Bernstein basis polynomials'.
A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm.
Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the Stone-Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval ''x'' ∈ [0, 1], became important in the form of Bézier curves.

Contents

Definition

Example

Properties

Approximating continuous functions

Proof

See also

References

Definition

The ''n'' + 1 'Bernstein basis polynomials' of degree ''n'' are defined as
:$b\_\{$
u,n}(x) = {n choose
u} x^{
u} (1-x)^{n-
u}, qquad
u=0,ldots,n.
where
:$\{n\; choose$
u}
is a binomial coefficient.
The Bernstein basis polynomials of degree ''n'' form a basis for the vector space $Pi\_n$ of polynomials of degree ''n''.
A linear combination of Bernstein basis polynomials
:$B(x)\; =\; sum\_\{$
u=0}^{n} eta_{
u} b_{
u,n}(x)
is called a 'Bernstein polynomial' or 'polynomial in Bernstein form' of degree ''n''. The coefficients β_ν are called 'Bernstein coefficients' or 'Bézier coefficients'.

Example

The first few Bernstein basis polynomials are
:$b\_\{0,0\}(x)\; =\; 1\; ,$
:$b\_\{0,1\}(x)\; =\; 1-x\; ,$
:$b\_\{1,1\}(x)\; =\; x\; ,$
:$b\_\{0,2\}(x)\; =\; (1-x)^2\; ,$
:$b\_\{1,2\}(x)\; =\; 2x(1-x)\; ,$
:$b\_\{2,2\}(x)\; =\; x^2\; .$

Properties

The Bernstein basis polynomials have the following properties:

★ $b\_\{$
u,n}(x) = 0, if ν < 0 or ν > ''n''

★ $b\_\{$
u,n}(0) = delta_{
u,0} and $b\_\{$
u,n}(1) = delta_{
u,n} where $delta$ is the Kronecker delta function.

★ $b\_\{$
u,n}(x) has a root with multiplicity ν at point ''x'' = 0 (note if ν is 0 there is no root at 0)

★ $b\_\{$
u,n}(x) has a root with multiplicity ''n'' − ν at point ''x'' = 1 (note if ν = ''n'' there is no root at 1)

★ $b\_\{$
u,n}(x) ≥ 0 for ''x'' in [0,1]

★ $b\_\{$
u,n}(1 - x) = b_{n-
u,n}(x)

★ If ν ≠ 0, then $b\_\{$
u,n}(x) has a unique local maximum on the interval [0,1] at ''x'' = ν/''n''. This maximum takes the value
::$$
u^{
u}n^{-n}(n-
u)^{n-
u}{nchoose
u}.

★ The Bernstein basis polynomials of degree ''n'' form a partition of unity:
::$sum\_\{$
u=0}^n b_{
u,n}(x) = sum_{
u=0}^n {n choose
u} x^{
u}(1-x)^{n-
u} = (x+(1-x))^n = 1.

★ A Bernstein polynomial can always be written as a linear combination of polynomials of higher degree.
::$b\_\{$
u,n-1}(x)=rac{n-
u}{n}b_{
u,n}(x)+rac{
u+1}{n}b_{
u+1,n}(x).

Approximating continuous functions

Let ''f''(''x'') be a continuous function on the interval [0, 1]. Consider the Bernstein polynomial
:$B\_n(f)(x)\; =\; sum\_\{$
u=0}^{n} fleft(rac{
u}{n}
ight) b_{
u,n}(x).
It can be shown that
:$lim\_\{n$
ightarrowinfty} B_n(f)(x)=f(x)
'uniformly on the interval [0, 1]'. This is a stronger statement than the proposition that the limit holds for each value of ''x'' separately; that would be pointwise convergence rather than uniform convergence. Specifically, the word ''uniformly'' signifies that
:$lim\_\{n$
ightarrowinfty} sup{,left|f(x)-B_n(f)(x)
ight|:0leq xleq 1,}=0.
Bernstein polynomials thus afford one way to prove the Stone-Weierstrass approximation theorem that every real-valued continuous function on a real interval [''a'',''b''] can be uniformly approximated by polynomial functions over 'R'.
A more general statement for a function with continuous k-th derivative is
:$|\; B\_n(f)^\{(k)\}\; |\_infty\; le\; \{(n)\_k\; over\; n^k\}\; |\; f^\{(k)\}\; |\_infty$ and $|f^\{(k)\}-\; B\_n(f)^\{(k)\}\; |\_infty$
ightarrow 0,
where additionally $\{(n)\_k\; over\; n^k\}=\; left(1-\{0\; over\; n\}$
ight)left(1-{1 over n}
ight) cdots left(1-{k-1 over n}
ight) is an eigenvalue of $B\_n$; the corresponding eigenfunction is a polynomial of degree k.

Proof

Suppose ''K'' is a random variable distributed as the number of successes in ''n'' independent Bernoulli trials with probability ''x'' of success on each trial; in other words, ''K'' has a binomial distribution with parameters ''n'' and ''x''. Then we have the expected value E(''K/n'') = ''x''.
Then the weak law of large numbers of probability theory tells us that
:
ight|>delta
ight)=0
for every $delta\; >\; 0$.
Because ''f'', being continuous on a closed bounded interval, must be uniformly continuous on that interval, we can infer a statement of the form
:$lim\_\{n$
ightarrowinfty} Pleft(left|f(K/n)-f(x)
ight|>arepsilon
ight)=0.
Consequently
:$lim\_\{n$
ightarrowinfty}
Pleft(left|f(K/n)-E(f(K/n))
ight|+left|E(f(K/n))-f(x)
ight|>arepsilon
ight)=0.
:$lim\_\{n$
ightarrowinfty}
Pleft(left|fleft(rac{K}{n}
ight)-Eleft(fleft(rac{K}{n}
ight)
ight)
ight|>arepsilon/2
ight)+Pleft(
left|Eleft(fleft(rac{K}{n}
ight)
ight)-f(x)
ight|>arepsilon/2
ight)=0.
And so the second probability above approaches 0 as ''n'' grows. But the second probability is either 0 or 1, since the only thing that is random is ''K'', and that appears ''within the scope of the expectation operator E''. Finally, observe that E(''f''(''K/n'')) is just the Bernstein polynomial ''B''_''n''(''f'',''x'').

See also

★ Bézier curve

★ Polynomial interpolation

★ Newton form

★ Lagrange form

References

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