Bernstein polynomial: Information from Answers.com

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

Bernstein polynomials approximating a curve

In the mathematical field 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.

1 Definition
2 Example
3 Properties
4 Approximating continuous functions
5 Proof
6 See also
7 References

Definition

The $\scriptstyle \left( n + 1 \right)$ Bernstein basis polynomials of degree $\scriptstyle n$ are defined as

$b_{\nu,n}(x) = {n \choose \nu} x^{\nu} \left( 1 - x \right)^{n - \nu}, \quad \nu = 0, \ldots, n.$

where $\scriptstyle {n \choose \nu}$ is a binomial coefficient.

The Bernstein basis polynomials of degree $\scriptstyle n$ form a basis for the vector space $\scriptstyle \Pi_n$ of polynomials of degree $\scriptstyle n$ .

A linear combination of Bernstein basis polynomials

$B(x) = \sum_{\nu=0}^{n} \beta_{\nu} b_{\nu,n}(x)$

is called a Bernstein polynomial or polynomial in Bernstein form of degree $\scriptstyle n$ . The coefficients $\scriptstyle \beta_\nu$ are called Bernstein coefficients or Bézier coefficients.

Example

The first few Bernstein basis polynomials are:

$\scriptstyle\ {b}_{0, 0} \left( {x} \right)\ =\ 1$	;
$\scriptstyle\ {b}_{0, 1} \left( {x} \right)\ =\ 1 - {x}$	;	$\scriptstyle\ {b}_{1 ,1} \left( {x} \right)\ =\ {x}$	;
$\scriptstyle\ {b}_{0, 2} \left( {x} \right)\ =\ {\left( 1 - {x} \right)}^{2}$	;	$\scriptstyle\ {b}_{1, 2} \left( {x} \right)\ =\ 2 {x} \left( 1 - {x} \right)$	;	$\scriptstyle\ {b}_{2, 2} \left( {x} \right)\ =\ {x}^{2}$	;
$\scriptstyle\ {b}_{0, 3} \left( {x} \right)\ =\ {\left( 1 - {x} \right)}^{3}$	;	$\scriptstyle\ {b}_{1, 3} \left( {x} \right)\ =\ 3 {x} {\left( 1 - {x} \right)}^{2}$	;	$\scriptstyle\ {b}_{2, 3} \left( {x} \right)\ =\ 3 {x}^{2} \left( 1 - {x} \right)$	;	$\scriptstyle\ {b}_{3, 3} \left( {x} \right)\ =\ {x}^{3}$	;
$\scriptstyle\ {b}_{0, 4} \left( {x} \right)\ =\ {\left( 1 - {x} \right)}^{4}$	;	$\scriptstyle\ {b}_{1, 4} \left( {x} \right)\ =\ 4 {x} {\left( 1 - {x} \right)}^{3}$	;	$\scriptstyle\ {b}_{2, 4} \left( {x} \right)\ =\ 6 {x}^{2} {\left( 1 - {x} \right)}^{2}$	;	$\scriptstyle\ {b}_{3, 4} \left( {x} \right)\ =\ 4 {x}^{3} \left( 1 - {x} \right)$	;	$\scriptstyle\ {b}_{4, 4} \left( {x} \right)\ =\ {x}^{4}$	...

Properties

The Bernstein basis polynomials have the following properties:

$\scriptstyle b_{\nu, n}(x)\ =\ 0$ , if $\scriptstyle \nu\ <\ 0$ or $\scriptstyle \nu\ >\ n$ .
$\scriptstyle b_{\nu, n}(0)\ =\ \delta_{\nu, 0}$ and $\scriptstyle b_{\nu, n}(1)\ =\ \delta_{\nu, n}$ where $\scriptstyle \delta$ is the Kronecker delta function.
$\scriptstyle b_{\nu, n}(x)$ has a root with multiplicity $\scriptstyle \nu$ at point $\scriptstyle x\ =\ 0$ (note: if $\scriptstyle \nu\ =\ 0$ , there is no root at 0).
$\scriptstyle b_{\nu, n}(x)$ has a root with multiplicity $\scriptstyle \left( n - \nu \right)$ at point $\scriptstyle x\ =\ 1$ (note: if $\scriptstyle \nu\ =\ n$ , there is no root at 1).
$\scriptstyle b_{\nu, n}(x)\ \ge\ 0$ for $\scriptstyle x\ \in\ [0,\ 1]$ .
$\scriptstyle b_{\nu, n}\left( 1 - x \right)\ =\ b_{n - \nu, n}(x)$ .

The derivative can be written as a combination of two polynomials of lower degree:

$b'_{\nu, n}(x) = n \left( b_{\nu - 1, n - 1}(x) - b_{\nu, n - 1}(x) \right).$

If $\scriptstyle n \ne 0$ , then $\scriptstyle b_{\nu, n}(x)$ has a unique local maximum on the interval $\scriptstyle [0,\ 1]$ at $\scriptstyle x\ =\ \frac{\nu}{n}$ . This maximum takes the value:

$\nu^\nu n^{-n} \left( n - \nu \right)^{n - \nu} {n \choose \nu}.$

The Bernstein basis polynomials of degree $\scriptstyle n$ form a partition of unity:

$\sum_{\nu = 0}^n b_{\nu, n}(x) = \sum_{\nu = 0}^n {n \choose \nu} x^\nu \left( 1 - x \right)^{n - \nu} = \left(x + \left( 1 - x \right) \right)^n = 1.$

A Bernstein polynomial can always be written as a linear combination of polynomials of higher degree:

$b_{\nu, n - 1}(x) = \frac{n - \nu}{n} b_{\nu, n}(x) + \frac{\nu + 1}{n} b_{\nu + 1, n}(x).$

Approximating continuous functions

Let $\scriptstyle f(x)$ be a continuous function on the interval $\scriptstyle \left[ 0, 1 \right]$ . Consider the Bernstein polynomial

$B_n(f)(x) = \sum_{\nu = 0}^{n} f\left( \frac{\nu}{n} \right) b_{\nu,n}(x).$

It can be shown that

$\lim_{n \to \infty}{ B_n(f)(x) } = f(x)$

uniformly on the interval $\scriptstyle \left[ 0, 1 \right]$ . This is a stronger statement than the proposition that the limit holds for each value of $\scriptstyle x$ separately; that would be pointwise convergence rather than uniform convergence. Specifically, the word uniformly signifies that

$\lim_{n \to \infty}{ \sup{ \left\{\, \left| f(x) - B_n(f)(x) \right| \,:\, 0 \leq x \leq 1 \,\right\} }} = 0$

Bernstein polynomials thus afford one way to prove the Stone-Weierstrass approximation theorem that every real-valued continuous function on a real interval $\scriptstyle \left[ a, b \right]$ can be uniformly approximated by polynomial functions over $\scriptstyle \R$ .

A more general statement for a function with continuous k-th derivative is

${\left\| B_n(f)^{(k)} \right\|}_\infty \le \frac{ (n)_k }{ n^k } {\left\| f^{(k)} \right\|}_\infty$ and ${\left\| f^{(k)}- B_n(f)^{(k)} \right\|}_\infty \to 0$

where additionally

$\frac{ (n)_k }{ n^k } = \left( 1 - \frac{0}{n} \right) \left( 1 - \frac{1}{n} \right) \cdots \left( 1 - \frac{k - 1}{n} \right)$

is an eigenvalue of $\scriptstyle B_n$ ; the corresponding eigenfunction is a polynomial of degree $\scriptstyle k$ .

Proof

Suppose $\scriptstyle K$ is a random variable distributed as the number of successes in $\scriptstyle n$ independent Bernoulli trials with probability $\scriptstyle x$ of success on each trial; in other words, $\scriptstyle K$ has a binomial distribution with parameters $\scriptstyle n$ and $\scriptstyle x$ . Then we have the expected value $\scriptstyle E\left( K/n \right) = x$ .

Then the weak law of large numbers of probability theory tells us that

$\lim_{n \to \infty}{ P\left( \left| \frac{K}{n} - x \right|>\delta \right) } = 0$

for every $\scriptstyle \delta > 0$ .

Because $\scriptstyle f$ , being continuous on a closed bounded interval, must be uniformly continuous on that interval, we can infer a statement of the form

$\lim_{n \to \infty}{ P\left( \left| f\left( \frac{K}{n} \right) - f\left( x \right) \right| > \varepsilon \right) } = 0$

Consequently

$\lim_{n \to \infty}{ P\left( \left| f\left( \frac{K}{n} \right) - E\left( f\left( \frac{K}{n} \right) \right) \right| + \left| E\left( f\left( \frac{K}{n} \right) \right) - f\left( x \right) \right| > \varepsilon \right) } = 0$

$\lim_{n \to \infty}{ P\left( \left| f\left( \frac{K}{n} \right) - E\left( f\left( \frac{K}{n} \right) \right) \right| > \frac{\varepsilon}{2} \right) + P\left( \left| E\left( f\left( \frac{K}{n} \right) \right) - f\left( x\right) \right| > \frac{\varepsilon}{2} \right) } = 0$

And so the second probability above approaches 0 as $\scriptstyle n$ grows. But the second probability is either 0 or 1, since the only thing that is random is $\scriptstyle K$ , and that appears within the scope of the expectation operator $\scriptstyle E$ . Finally, observe that $\scriptstyle E\left( f\left( K/n \right) \right)$ is just the Bernstein polynomial $\scriptstyle B_n(f, x)$ .

References

Weisstein, Eric W., "Bernstein Polynomial" from MathWorld.
This article incorporates material from properties of Bernstein polynomial on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
From Bézier to Bernstein
BERNSTEIN POLYNOMIALS by Kenneth I. Joy

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Bernstein polynomial

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