Bernstein polynomial

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Bernstein polynomials approximating a curve

In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials. The idea is named after Sergei Natanovich Bernstein.

A numerically 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 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.

Bernstein basis polynomials for 4th degree curve blending

Definition[edit]

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. A linear combination of Bernstein basis polynomials

is called a Bernstein polynomial or polynomial in Bernstein form of degree n.[1] The coefficients are called Bernstein coefficients or Bézier coefficients.

The first few Bernstein basis polynomials from above in monomial form are:

Properties[edit]

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[2]
  • 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[3]

Approximating continuous functions[edit]

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 [ab] 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.

Probabilistic proof[edit]

This proof follows Bernstein's original proof of 1912.[8] See also Feller (1966) or Koralov & Sinai (2007).[9][10]

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 1n K, equal to 1n 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

Elementary proof[edit]

The probabilistic proof can also be rephrased in an elementary way, using the underlying probabilistic ideas but proceeding by direct verification:[11][12][13][14][15]

The following identities can be verified:

(1)

("probability")

(2)

("mean")

(3)

("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 2M times

(Chebyshev's inequality)

It follows that the polynomials fn tend to f uniformly.

Generalizations to higher dimension[edit]

Bernstein polynomials can be generalized to k dimensions. The resulting polynomials have the form Pi1(x1) Pi2(x2) ... Pik(xk).[16] 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. [17]

See also[edit]

Notes[edit]

  1. ^ a b Lorentz 1953
  2. ^ Mathar, R. J. (2018). "Orthogonal basis function over the unit circle with the minimax property". Appendix B. arXiv:1802.09518.
  3. ^ Rababah, Abedallah (2003). "Transformation of Chebyshev-Bernstein Polynomial Basis". Comp. Meth. Appl. Math. 3 (4): 608–622. doi:10.2478/cmam-2003-0038.
  4. ^ Natanson (1964) p. 6
  5. ^ Feller 1966
  6. ^ Beals 2004
  7. ^ Natanson (1964) p. 3
  8. ^ Bernstein 1912
  9. ^ Koralov, L.; Sinai, Y. (2007). ""Probabilistic proof of the Weierstrass theorem"". Theory of probability and random processes (2nd ed.). Springer. p. 29.
  10. ^ Feller 1966
  11. ^ Lorentz 1953, pp. 5–6
  12. ^ Beals 2004
  13. ^ Goldberg 1964
  14. ^ Akhiezer 1956
  15. ^ Burkill 1959
  16. ^ Lorentz 1953
  17. ^ Hildebrandt, T. H.; Schoenberg, I. J. (1933), "On linear functional operations and the moment problem for a finite interval in one or several dimensions", Annals of Mathematics, 34: 327

References[edit]

External links[edit]