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 numerical 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 [0,1], became important in the form of Bézier curves.

Definition

The n+1 Bernstein basis polynomials of degree n are defined as

The Bernstein basis polynomials of degree n form a basis for the vector space Π_n of polynomials of degree n.

A linear combination of Bernstein basis polynomials

is called Bernstein polynomial or polynomial in Bernstein form of degree n. The coefficients β_ν are called Bernstein coefficients or Bézier coefficients.

Notes

The Bernstein basis polynomials have the following properties

• b_ν,n(x) has a root with multiplicity ν at point x = 0
• b_ν,n(x) has a root with multiplicity n − ν at point x = 1
• b_ν,n(x) ≥ 0 if x in [0,1]
• b_ν,n(x) has a global maximum at x = ν/n
• b’_ν,n(x) = n [b_ν-1,n-1(x) - b_ν,n-1(x)]
• b_ν,n(x) = 0, if ν < 0 or ν > n

The Bernstein basis polynomials of degree n form a partition of unity

Example

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.

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

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

Consequently

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