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.
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 polynomialsedit
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:
Propertiesedit
The Bernstein basis polynomials have the following properties:
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.
Probabilistic proofedit
This proof follows Bernstein's original proof of 1912.[8] See also Feller (1966) or Koralov & Sinai (2007).[9][5]
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⁄nK, equal to 1⁄nx(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 proofedit
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:
("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
It follows that the polynomials fn tend to f uniformly.
Generalizations to higher dimensionedit
Bernstein polynomials can be generalized to k dimensions – the resulting polynomials have the form Bi1(x1) Bi2(x2) ... Bik(xk).[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.
[14]
^Koralov, L.; Sinai, Y. (2007). ""Probabilistic proof of the Weierstrass theorem"". Theory of probability and random processes (2nd ed.). Springer. p. 29.
Bernstein, S. (1912), "Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités (Proof of the theorem of Weierstrass based on the calculus of probabilities)" (PDF), Comm. Kharkov Math. Soc., 13: 1–2, English translation
Akhiezer, N. I. (1956), Theory of approximation (in Russian), translated by Charles J. Hyman, Frederick Ungar, pp. 30–31, Russian edition first published in 1940
Goldberg, Richard R. (1964), Methods of real analysis, John Wiley & Sons, pp. 263–265
Caglar, Hakan; Akansu, Ali N. (July 1993). "A generalized parametric PR-QMF design technique based on Bernstein polynomial approximation". IEEE Transactions on Signal Processing. 41 (7): 2314–2321. Bibcode:1993ITSP...41.2314C. doi:10.1109/78.224242. Zbl 0825.93863.
Natanson, I.P. (1964). Constructive function theory. Volume I: Uniform approximation. Translated by Alexis N. Obolensky. New York: Frederick Ungar. MR 0196340. Zbl 0133.31101.
Feller, William (1966), An introduction to probability theory and its applications, Vol, II, John Wiley & Sons, pp. 149–150, 218–222
Kac, Mark (1938). "Une remarque sur les polynomes de M. S. Bernstein". Studia Mathematica. 7: 49–51. doi:10.4064/sm-7-1-49-51.
Kelisky, Richard Paul; Rivlin, Theodore Joseph (1967). "Iteratives of Bernstein Polynomials". Pacific Journal of Mathematics. 21 (3): 511. doi:10.2140/pjm.1967.21.511.
Stark, E. L. (1981). "Bernstein Polynome, 1912-1955". In Butzer, P.L. (ed.). ISNM60. pp. 443–461. doi:10.1007/978-3-0348-9369-5_40. ISBN 978-3-0348-9369-5.
Joy, Kenneth I. (2000). "Bernstein Polynomials" (PDF). Archived from the original (PDF) on 2012-02-20. Retrieved 2009-02-28. from University of California, Davis. Note the error in the summation limits in the first formula on page 9.
Idrees Bhatti, M.; Bracken, P. (2007). "Solutions of differential equations in a Bernstein Polynomial basis". J. Comput. Appl. Math. 205 (1): 272–280. Bibcode:2007JCoAM.205..272I. doi:10.1016/j.cam.2006.05.002.
Acikgoz, Mehmet; Araci, Serkan (2010). "On the generating function for Bernstein Polynomials". AIP Conf. Proc. AIP Conference Proceedings. 1281 (1): 1141. Bibcode:2010AIPC.1281.1141A. doi:10.1063/1.3497855.
Doha, E. H.; Bhrawy, A. H.; Saker, M. A. (2011). "Integrals of Bernstein polynomials: An application for the solution of high even-order differential equations". Appl. Math. Lett. 24 (4): 559–565. doi:10.1016/j.aml.2010.11.013.
Farouki, Rida T. (2012). "The Bernstein polynomial basis: a centennial retrospective". Comp. Aid. Geom. Des. 29 (6): 379–419. doi:10.1016/j.cagd.2012.03.001.
Chen, Xiaoyan; Tan, Jieqing; Liu, Zhi; Xie, Jin (2017). "Approximations of functions by a new family of generalized Bernstein operators". J. Math. Ann. Applic. 450: 244–261. doi:10.1016/j.jmaa.2016.12.075.
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