Bernstein polynomial

Bernstein polynomial
In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial expressed as a linear combination of Bernstein basis polynomials. The idea is named after mathematician Sergei Natanovich Bernstein.
Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the 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.
A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm.
Definition
Bernstein basis polynomials
The n+1 Bernstein basis polynomials of degree n are defined as
where is a binomial coefficient.
So, for example,
The first few Bernstein basis polynomials for blending 1, 2, 3 or 4 values together are:
The Bernstein basis polynomials of degree n form a basis for the vector space of polynomials of degree at most n with real coefficients.
Bernstein polynomials
A linear combination of Bernstein basis polynomials
is called a Bernstein polynomial or polynomial in Bernstein form of degree n. The coefficients are called Bernstein coefficients or Bézier coefficients.
The first few Bernstein basis polynomials from above in monomial form are:
Properties
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 k-th derivative at 0:
The k-th derivative at 1:
The transformation of the Bernstein polynomial to monomials is and by the inverse binomial transformation, the reverse transformation is
The indefinite integral is given by
The definite integral is constant for a given n:
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 , treating as constant, then substituting the value , it can be shown that
Similarly the second -derivative of , with again then substituted , shows that
A Bernstein polynomial can always be written as a linear combination of polynomials of higher degree:
The expansion of the Chebyshev Polynomials of the First Kind into the Bernstein basis is
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].
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
where additionally
is an eigenvalue of Bn; the corresponding eigenfunction is a polynomial of degree k.
Probabilistic proof
This proof follows Bernstein's original proof of 1912. See also Feller (1966) or Koralov & Sinai (2007).
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
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 K, equal to x(1−x), is bounded from above by 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
Elementary proof
The probabilistic proof can also be rephrased in an elementary way, using the underlying probabilistic ideas but proceeding by direct verification:
The following identities can be verified:
("probability")
("mean")
("variance")
In fact, by the binomial theorem
and this equation can be applied twice to . The identities (1), (2), and (3) follow easily using the substitution .
Within these three identities, use the above basis polynomial notation
and let
Thus, by identity (1)
so that
Since f is uniformly continuous, given , there is a such that whenever
. Moreover, by continuity, . But then
The first sum is less than ε. On the other hand, by identity (3) above, and since , the second sum is bounded by times
(Chebyshev's inequality)
It follows that the polynomials fn tend to f uniformly.
Generalizations to higher dimension
Bernstein polynomials can be generalized to dimensions – the resulting polynomials have the form . In the simplest case only products of the unit interval are considered; but, using affine transformations of the line, Bernstein polynomials can also be defined for products . For a continuous function on the -fold product of the unit interval, the proof that can be uniformly approximated by
is a straightforward extension of Bernstein's proof in one dimension.
See also
Polynomial interpolation
Newton form
Lagrange form
Binomial QMF (also known as Daubechies wavelet)
Notes
References
, English translation
, Russian edition first published in 1940
External links
from University of California, Davis. Note the error in the summation limits in the first formula on page 9.
Feature Column from American Mathematical Society
Numerical analysis
Polynomials
Articles containing proofs