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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.
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
where is a binomial coefficient. So, for example,
is called a Bernstein polynomial or polynomial in Bernstein form of degree n. The coefficients are called Bernstein coefficients or Bézier coefficients.
The Bernstein basis polynomials have the following properties:
- , if or .
- for .
- and where is the Kronecker delta function:
- has a root with multiplicity at point (note: if , there is no root at 0).
- has a root with multiplicity at point (note: if , there is no root at 1).
- The derivative can be written as a combination of two polynomials of lower degree:
- The transformation of the Bernstein polynomial to monomials is
- The indefinite integral is given by:
- The definite integral is constant for a given
- If , then has a unique local maximum on the interval at . This maximum takes the value:
- The Bernstein basis polynomials of degree form a partition of unity:
- By taking the first derivative of where , it can be shown that
- The second derivative of where can be used to show
- A Bernstein polynomial can always be written as a linear combination of polynomials of higher degree:
Approximating continuous functions
Let ƒ be a continuous function on the interval [0, 1]. Consider the Bernstein polynomial
It can be shown that
uniformly on the interval [0, 1]. The word uniformly signifies that the polynomial converges on the entire interval [0, 1] at the same rate (or better). This is a more stringent form of convergence than pointwise convergence, which only requires that the limit is achieved at each value of x on [0, 1], with (possibly) separate rates at each point. Specifically, uniform convergence assures that
Bernstein polynomials thus provide one way to prove the Weierstrass approximation theorem that every real-valued continuous function on a real interval [a, b] can be uniformly approximated by polynomial functions over .
A more general statement for a function with continuous kth derivative is
is an eigenvalue of Bn; the corresponding eigenfunction is a polynomial of degree k.
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 and
for every δ > 0. Moreover, this relation holds uniformly in x, which can be seen from its proof via Chebyshev's inequality, taking into account that the variance of 1⁄n K, equal to 1⁄n x(1−x), is bounded from above by 1⁄(4n) irrespective of x.
Because ƒ, being continuous on a closed bounded interval, must be uniformly continuous on that interval, one infers a statement of the form
uniformly in x. Taking into account that ƒ is bounded (on the given interval) one gets for the expectation
uniformly in x. To this end one splits the sum for the expectation in two parts. On one part the difference does not exceed ε; this part cannot contribute more than ε. On the other part the difference exceeds ε, but does not exceed 2M, where M is an upper bound for |ƒ(x)|; this part cannot contribute more than 2M times the small probability that the difference exceeds ε.
Finally, one observes that the absolute value of the difference between expectations never exceeds the expectation of the absolute value of the difference, and
See for instance Koralov & Sinai (2007).
- G. G. Lorentz (1953) Bernstein Polynomials, University of Toronto Press
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- Natanson (1964) p. 3
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- This article incorporates material from properties of Bernstein polynomial on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.