Bernstein polynomial

The idea is named after mathematician Sergei Natanovich Bernstein.

With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0, 1], became important in the form of Bézier curves.

A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm.

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

The first few Bernstein basis polynomials from above in monomial form are: The Bernstein basis polynomials have the following properties: 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].

[4][1][5][6] 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

[7] A more general statement for a function with continuous kth derivative is where additionally is an eigenvalue of Bn; the corresponding eigenfunction is a polynomial of degree k. This proof follows Bernstein's original proof of 1912.

[9][5] We will first give intuition for Bernstein's original proof.

This consideration renders the approximation theorem intuitive, given that polynomials should be flexible enough to match (or nearly match) a finite number of pairs

The probabilistic proof below simply provides a constructive method to create a polynomial which is approximately equal to

on such a point lattice, given that "smoothing out" a function is not always trivial.

Taking the expectation of a random variable with a simple distribution is a common way to smooth.

Here, we take advantage of the fact that Bernstein polynomials look like Binomial expectations.

We split the interval into a lattice of n discrete values.

The proof below illustrates that this achieves a uniform approximation of f. The crux of the proof is to (1) justify replacing an arbitrary point with a binomially chosen lattice point by concentration properties of a Binomial distribution, and (2) justify the inference from

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.

and By the weak law of large numbers of probability theory, 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 for each

Taking into account that ƒ is bounded (on the given interval) one finds that uniformly in x.

Thus, this part of the expectation contributes no more than 2M times ε.

Finally, one observes that the absolute value of the difference between expectations never exceeds the expectation of the absolute value of the difference, a consequence of Holder's Inequality.

is a polynomial in x (the subscript reminding us that x controls the distribution of K).

Indeed it is: In the above proof, recall that convergence in each limit involving f depends on the uniform continuity of f, which implies a rate of convergence dependent on f 's modulus of continuity

Thus, the approximation only holds uniformly across x for a fixed f, but one can readily extend the proof to uniformly approximate a set of functions with a set of Bernstein polynomials in the context of equicontinuity.

The probabilistic proof can also be rephrased in an elementary way, using the underlying probabilistic ideas but proceeding by direct verification:[10][6][11][12][13] The following identities can be verified: In fact, by the binomial theorem

times It follows that the polynomials fn tend to f uniformly.

[1] In the simplest case only products of the unit interval [0,1] are considered; but, using affine transformations of the line, Bernstein polynomials can also be defined for products [a1, b1] × [a2, b2] × ... × [ak, bk].

For a continuous function f on the k-fold product of the unit interval, the proof that f(x1, x2, ... , xk) can be uniformly approximated by is a straightforward extension of Bernstein's proof in one dimension.