Science Fair Project Encyclopedia
- 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.
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 [0,1], became important in the form of Bézier curves.
The n + 1 Bernstein basis polynomials of degree n are defined as
A linear combination of Bernstein basis polynomials
is called a Bernstein polynomial or polynomial in Bernstein form of degree n. The coefficients βν are called Bernstein coefficients or Bézier coefficients.
The Bernstein basis polynomials have the following properties:
- bν,n(x) has a root with multiplicity ν at point x = 0
- bν,n(x) has a root with multiplicity n − ν at point x = 1
- bν,n(x) ≥ 0 if x in [0,1]
- bν,n(x) has a global maximum at x = ν/n
- b’ν,n(x) = n [bν-1,n-1(x) - bν,n-1(x)]
- bν,n(x) = 0, if ν < 0 or ν > n
The Bernstein basis polynomials of degree n form a partition of unity:
The first few Bernstein basis polynomials are
Approximating continuous functions
Let f(x) be a continuous function on the interval [0, 1]. Consider the Bernstein polynomial
It can be shown that
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
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.
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.
Because f, being continuous on a closed bounded interval, must be uniformly continuous on that interval, we can infer a statement of the form
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 Bn(f,x).
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details