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{{for|the Bernstein polynomial in [[D-module]] theory|Bernstein&ndash;Sato polynomial}}
{{more footnotes|date=June 2016}}[[Image:Bernstein Approximation.gif|thumb|right|Bernstein polynomials approximating a curve]]
In the [[mathematics|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'''.

A [[numerical stability|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 [[Stone–Weierstrass theorem|Stone–Weierstrass approximation theorem]]. With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0,&nbsp;1], became important in the form of [[Bézier curve]]s.

==Definition==
[[File:Bernstein Polynomials.svg|thumb|Bernstein basis polynomials for 4th degree curve blending]]
The ''n''{{Hair space}}+1 '''Bernstein basis polynomials''' of degree ''n'' are defined as

: <math>b_{\nu,n}(x) = \binom{n}{\nu} x^{\nu} \left( 1 - x \right)^{n - \nu}, \quad \nu = 0, \ldots, n,</math>

where <math>\tbinom{n}{\nu}</math> is a [[binomial coefficient]].  So, for example, <math>b_{2,5}(x) = \tbinom{5}{2}x^2(1-x)^3 = 10x^2(1-x)^3.</math>

The first few Bernstein basis polynomials for blending 1, 2, 3 or 4 values together are:
: <math>
\begin{align}
b_{0,0}(x) & = 1, \\
b_{0,1}(x) & = 1 - x, & b_{1,1}(x) & = x \\
b_{0,2}(x) & = (1 - x)^2, & b_{1,2}(x) & = 2x(1 - x), & b_{2,2}(x) & = x^2 \\
b_{0,3}(x) & = (1 - x)^3, & b_{1,3}(x) & = 3x(1 - x)^2, & b_{2,3}(x) & = 3x^2(1 - x), & b_{3,3}(x) & = x^3
\end{align}
</math>
:

The Bernstein basis polynomials of degree ''n'' form a [[basis (linear algebra)|basis]] for the [[vector space]] Π<sub>''n''</sub> of polynomials of degree at most&nbsp;''n'' with real coefficients.  A linear combination of Bernstein basis polynomials

:<math>B_n(x) = \sum_{\nu=0}^{n} \beta_{\nu} b_{\nu,n}(x)</math>

is called a '''Bernstein polynomial''' or '''polynomial in Bernstein form''' of degree&nbsp;''n''.<ref>G. G. Lorentz (1953) ''Bernstein Polynomials'', [[University of Toronto Press]]</ref>  The coefficients <math>\beta_\nu</math> are called '''Bernstein coefficients''' or '''Bézier coefficients'''.

The first few Berntein basis polynomials from above in monomial form are:
: <math>
\begin{align}
b_{0,0}(x) & = 1, \\
b_{0,1}(x) & = 1 - 1x, & b_{1,1}(x) & = 0 + 1x \\
b_{0,2}(x) & = 1 - 2x + 1x^2, & b_{1,2}(x) & = 0 + 2x - 2x^2, & b_{2,2}(x) & = 0 + 0x + 1x^2 \\
b_{0,3}(x) & = 1 - 3x + 3x^2 - x^3, & b_{1,3}(x) & = 0 + 3x - 6x^2 + 3x^3, & b_{2,3}(x) & = 0 + 0x + 3x^2 - 3x^3, & b_{3,3}(x) & = 0 + 0x + 0x^2 + 1x^3
\end{align}
</math>
:

==Properties==
The Bernstein basis polynomials have the following properties:
* <math>b_{\nu, n}(x) = 0</math>, if <math>\nu < 0</math> or <math>\nu > n.</math>

* <math>b_{\nu, n}(x) \ge 0</math> for <math>x \in [0,\ 1].</math>

* <math>b_{\nu, n}\left( 1 - x \right) = b_{n - \nu, n}(x).</math>

* <math>b_{\nu, n}(0) = \delta_{\nu, 0}</math> and <math>b_{\nu, n}(1) = \delta_{\nu, n}</math> where <math>\delta</math> is the [[Kronecker delta]] function: <math>\delta_{ij} = \begin{cases}
0 &\text{if } i \neq j,   \\
1 &\text{if } i=j.   \end{cases}</math>

* <math>b_{\nu, n}(x)</math> has a root with multiplicity <math>\nu</math> at point <math>x = 0</math> (note: if <math>\nu = 0</math>, there is no root at 0).

* <math>b_{\nu, n}(x)</math> has a root with multiplicity <math>\left( n - \nu \right)</math> at point <math>x = 1</math> (note: if <math>\nu = n</math>, there is no root at 1).

* The [[derivative]] can be written as a combination of two polynomials of lower degree:
*: <math>b'_{\nu, n}(x) = n \left( b_{\nu - 1, n - 1}(x) - b_{\nu, n - 1}(x) \right).</math>

* The transformation of the Bernstein polynomial to monomials is
*: <math>b_{\nu,n}(x) = \binom{n}{\nu}\sum_{k=0}^{n-\nu} \binom{n-\nu}{k}(-1)^{n-\nu-k} x^{\nu+k} = \sum_{\ell=\nu}^n \binom{n}{\ell}\binom{\ell}{\nu}(-1)^{\ell-\nu}x^\ell,</math>
:and by the [[Binomial transform|inverse binomial transformation]], the reverse transformation is<ref>{{cite arxiv |eprint=1802.09518 |first=R. J. |last=Mathar |year=2018 |title=Orthogonal basis function over the unit circle with the minimax property |at=Appendix&nbsp;B}}</ref>
: <math>x^k=\sum_{i=0}^{n-k}\binom{n-k}{i}\frac{1}{\binom{n}{i}}b_{n-i,n}(x) = \frac{1}{\binom{n}{k}} \sum_{j=k}^n \binom{j}{k}b_{j,n}(x).</math>

* The indefinite [[integral]] is given by
*: <math>\int b_{\nu, n}(x)dx = \frac{1}{n+1} \sum_{j=\nu+1}^{n+1} b_{j, n+1}(x).</math> 

* The definite integral is constant for a given ''n'':
*: <math>\int_{0}^{1}b_{\nu, n}(x)dx = \frac{1}{n+1} \quad\ \, \text{for all } \nu = 0,1, \dots, n.</math>

* If <math>n \ne 0</math>, then <math>b_{\nu, n}(x)</math> has a unique local maximum on the interval <math>[0,\ 1]</math> at <math>x = \frac{\nu}{n}</math>. This maximum takes the value
*: <math>\nu^\nu n^{-n} \left( n - \nu \right)^{n - \nu} {n \choose \nu}.</math>

* The Bernstein basis polynomials of degree <math>n</math> form a [[partition of unity]]:
*: <math>\sum_{\nu = 0}^n b_{\nu, n}(x) = \sum_{\nu = 0}^n {n \choose \nu} x^\nu \left(1 - x\right)^{n - \nu} = \left(x + \left( 1 - x \right) \right)^n = 1.</math>

* By taking the first derivative of <math>(x+y)^n</math> where <math>y = 1-x</math>, it can be shown that
*: <math>\sum_{\nu=0}^{n}\nu b_{\nu, n}(x) = nx.</math>

* The second derivative of <math>(x+y)^n</math> where <math>y = 1-x</math> can be used to show
*: <math>\sum_{\nu=1}^{n}\nu(\nu-1) b_{\nu, n}(x) = n(n-1)x^2.</math>

* A Bernstein polynomial can always be written as a linear combination of polynomials of higher degree:
*: <math>b_{\nu, n - 1}(x) = \frac{n - \nu}{n} b_{\nu, n}(x) + \frac{\nu + 1}{n} b_{\nu + 1, n}(x).</math>

==Approximating continuous functions==
Let ''&fnof;'' be a [[continuous function]] on the interval [0,&nbsp;1]. Consider the Bernstein polynomial
:<math>B_n(f)(x) = \sum_{\nu = 0}^n f\left( \frac{\nu}{n} \right) b_{\nu,n}(x).</math>

It can be shown that
:<math>\lim_{n \to \infty}{ B_n(f) } = f </math>

[[uniform convergence|uniformly]] on the interval&nbsp;[0,&nbsp;1].<ref name=Nat6>Natanson (1964) p.&nbsp;6</ref> The word [[uniform convergence|''uniformly'']] signifies that the polynomial converges on the entire interval [0,&nbsp;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,&nbsp;1], with (possibly) separate rates at each point. Specifically, [[uniform convergence]] assures that
:<math>\lim_{n \to \infty} \sup \left\{\, \left| f(x) - B_n(f)(x) \right| \,:\, 0 \leq x \leq 1 \,\right\} = 0.</math>

Bernstein polynomials thus provide one way to prove the [[Stone&ndash;Weierstrass theorem#Weierstrass approximation theorem|Weierstrass approximation theorem]] that every real-valued continuous function on a real interval [''a'',&nbsp;''b''] can be uniformly approximated by polynomial functions over&nbsp;<math>\mathbb R</math>.<ref name=Nat3>Natanson (1964) p.&nbsp;3</ref>

A more general statement for a function with continuous ''k''<sup>th</sup> derivative is
:<math>{\left\| B_n(f)^{(k)} \right\|}_\infty \le \frac{ (n)_k }{ n^k } \left\| f^{(k)} \right\|_\infty \quad\ \text{and} \quad\ \left\| f^{(k)}- B_n(f)^{(k)} \right\|_\infty \to 0,</math>
where additionally
:<math>\frac{ (n)_k }{ n^k } = \left( 1 - \frac{0}{n} \right) \left( 1 - \frac{1}{n} \right) \cdots \left( 1 - \frac{k - 1}{n} \right)</math>
is an [[eigenvalue]] of ''B''<sub>''n''</sub>; the corresponding eigenfunction is a polynomial of degree&nbsp;''k''.

===Proof===
Suppose ''K'' is a [[random variable]] distributed as the number of successes in ''n'' independent [[Bernoulli trial]]s with probability ''x'' of success on each trial; in other words, ''K'' has a [[binomial distribution]] with parameters ''n'' and&nbsp;''x''. Then we have the [[expected value]] <math>\operatorname{\mathcal E}\left[\frac{K}{n}\right] = x\ </math> and
:<math>p(K) = {n \choose K} x^{K} \left( 1 - x \right)^{n - K} = b_{K,n}(x)</math>

By the [[law of large numbers|weak law of large numbers]] of [[probability theory]],
:<math>\lim_{n \to \infty}{ P\left( \left| \frac{K}{n} - x \right|>\delta \right) } = 0</math>

for every ''&delta;''&nbsp;>&nbsp;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 {{frac|1|''n''}}&nbsp;''K'', equal to {{frac|1|''n''}}&nbsp;''x''(1&minus;''x''), is bounded from above by {{frac|1|(4''n'')}} irrespective of ''x''.

Because ''&fnof;'', being continuous on a closed bounded interval, must be [[uniform continuity|uniformly continuous]] on that interval, one infers a statement of the form
:<math>\lim_{n \to \infty}{ P\left( \left| f\left( \frac{K}{n} \right) - f\left( x \right) \right| > \varepsilon \right) } = 0</math>

uniformly in ''x''. Taking into account that ''ƒ'' is bounded (on the given interval) one gets for the expectation
: <math>\lim_{n \to \infty}{ \operatorname{\mathcal E}\left( \left| f\left( \frac{K}{n} \right) - f\left( x \right) \right| \right) } = 0</math>
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 2''M'', where ''M'' is an upper bound for |''&fnof;''(x)|; this part cannot contribute more than 2''M'' 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 
:<math>\operatorname{\mathcal E}\left[f\left(\frac{K}{n}\right)\right] = \sum_{K=0}^n f\left(\frac{K}{n}\right) p(K) = \sum_{K=0}^n f\left(\frac{K}{n}\right) b_{K,n}(x) = B_n(f)(x)</math>

See for instance Koralov & Sinai (2007).<ref>{{cite book |first1=L. |last1=Koralov |first2=Y. |last2=Sinai |title=Theory of probability and random processes |edition=2nd |publisher=Springer |year=2007 |page=29 |chapter="Probabilistic proof of the Weierstrass theorem"}}</ref>

==See also==
*[[Polynomial interpolation]]
*[[Newton polynomial|Newton form]]
*[[Lagrange polynomial|Lagrange form]]
*[[Binomial QMF]]

==Notes==
<references />

==References==
* {{cite journal | last1=Caglar | first1=Hakan | last2=Akansu | first2=Ali N. | title=A generalized parametric PR-QMF design technique based on Bernstein polynomial approximation | zbl=0825.93863 | journal=IEEE Transactions on Signal Processing | volume=41 | number=7 | pages=2314–2321 | date=July 1993 | doi=10.1109/78.224242}}
* {{springer|title=Bernstein polynomials|id=B/b015730|last=Korovkin|first=P.P.}}
* {{cite book | last=Natanson | first=I.P. | authorlink=Isidor Natanson | title=Constructive function theory. Volume I: Uniform approximation | translator=Alexis N. Obolensky | zbl=0133.31101 | mr=0196340 | location=New York | publisher=Frederick Ungar | year=1964 }}

==External links==
* {{cite journal|first1=Mark|last1=Kac| authorlink=Mark Kac|title=Une remarque sur les polynomes de M. S. Bernstein |journal = [[Studia Mathematica]] | year =1938| volume =7 | pages=49–51| doi=10.4064/sm-7-1-49-51}}
* {{cite journal|first1=Richard Paul | last1=Kelisky | first2=Theodore Joseph | last2=Rivlin
  |title=Iteratives of Bernstein Polynomials|year=1967 | journal=[[Pacific Journal of Mathematics]] | volume=21 | number=3 
  | page =511|doi=10.2130/pjm.1967.21.511}}
* {{cite book|first1=E. L. | last1=Stark|chapter = Bernstein Polynome, 1912-1955 | year=1981| doi=10.1007/978-3-0348-9-369-5_40|pages=443–461|editor-first=P.L. | editor-last=Butzer
  |title=ISNM60|isbn=978-3-0348-9369-5}}
* {{cite journal|first1=Sonia |last1=Petrone |authorlink= Sonia Petrone | title=Random Bernstein polynomials | journal=Scand. J. Stat. 
  | year=1999|volume=26|number=3|pages=373–393|doi=10.1111/1467-9469.00155}}
* {{cite journal|first1=Halil |last1=Oruc |first2=Geoerge M. |last2=Phillips|title= A generalization of the Bernstein Polynomials
  | year=1999 | journal=[[Edinburgh_Mathematical_Society#Journals|Proceedings of the Edinburgh Mathematical Society]]|volume=42 |pages=403–413 |doi=10.1017/S0013091500020332}}
* {{cite web|first1=Kenneth I. |last1= Joy |year=2000 |url =http://www.idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials.pdf |title=Bernstein Polynomials}} from [[University of California, Davis]]. Note the error in the summation limits in the first formula on page 9.
* {{cite journal|first1=M.|last1=Idrees Bhatti |first2=P. |last2=Bracken | title=Solutions of differential equations in a Bernstein Polynomial basis|
   doi=10.1016/j.cam.2006.05.002|year=2007|journal=J. Comput. Appl. Math.|pages=272–280|volume=205}}
* {{cite web|first1=Bill|last1=Casselman|authorlink=Bill Casselman (mathematician)|url=http://www.ams.org/featurecolumn/archive/bezier.html| title= From Bézier to Bernstein|year=2008}} Feature Column from [[American Mathematical Society]]
* {{cite journal|first1=Mehmet | last1=Acikgoz |first2=Serkan |last2=Araci|title=On the generating function for Bernstein Polynomials | year=2010 | journal = AIP Conf. Proc. | doi=10.1063/1.3497855|volume=1281 |page=1141}}
* {{cite journal|first1=E. H. |last1=Doha|first2=A. H.|last2=Bhrawy |first3=M. A. |last3=Saker
  |title=Integrals of Bernstein polynomials: An application for the solution of high even-order differential equations
  |doi=10.1016/j.aml.2010.11.013| year=2011 | journal=Appl. Math. Lett.|volume=24 | pages=559–565}}
* {{cite journal|first1=Rida T.|last1=Farouki | title=The Bernstein polynomial basis: a centennial retrospective | year=2012|journal= Comp. Aid. Geom. Des.|volume=29|pages=379–419|doi=10.1016/j.cagd.2012.03.001}}
* {{cite journal|first1=Xiaoyan|last1=Chen|first2=Jieqing|last2=Tan|first3=Zhi|last3=Liu|first4=Jin|last4=Xie| 
  title=Approximations of functions by a new family of generalized Bernstein operators|journal=J. Math. Ann. Applic. | year=2017 | volume=450 | pages=244–261 | doi=10.1016/j.jmaa.2016.12.075}}
* {{mathworld|urlname=BernsteinPolynomial|title=Bernstein Polynomial}}
* {{PlanetMath attribution|urlname=BernsteinPolynomial|title=properties of Bernstein polynomial}}

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