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

${\displaystyle b_{\nu ,n}(x)\mathrel {:} \mathrel {=} {\binom {n}{\nu }}x^{\nu }\left(1-x\right)^{n-\nu },\quad \nu =0,\ldots ,n,}$

where ${\displaystyle {\tbinom {n}{\nu }}}$ is a binomial coefficient.

So, for example, ${\displaystyle b_{2,5}(x)={\tbinom {5}{2}}x^{2}(1-x)^{3}=10x^{2}(1-x)^{3}.}$

The first few Bernstein basis polynomials for blending 1, 2, 3 or 4 values together are:

${\displaystyle {\begin{aligned}b_{0,0}(x)&=1,\\b_{0,1}(x)&=1-x,&b_{1,1}(x)&=x\\b_{0,2}(x)&=(1-x)^{2},&b_{1,2}(x)&=2x(1-x),&b_{2,2}(x)&=x^{2}\\b_{0,3}(x)&=(1-x)^{3},&b_{1,3}(x)&=3x(1-x)^{2},&b_{2,3}(x)&=3x^{2}(1-x),&b_{3,3}(x)&=x^{3}\end{aligned}}}$

The Bernstein basis polynomials of degree n form a basis for the vector space ${\displaystyle \Pi _{n}}$ of polynomials of degree at most n with real coefficients.

Bernstein polynomials

A linear combination of Bernstein basis polynomials

${\displaystyle B_{n}(x)\mathrel {:} \mathrel {=} \sum _{\nu =0}^{n}\beta _{\nu }b_{\nu ,n}(x)}$

is called a Bernstein polynomial or polynomial in Bernstein form of degree n.^[1] The coefficients ${\displaystyle \beta _{\nu }}$ are called Bernstein coefficients or Bézier coefficients.

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

${\displaystyle {\begin{aligned}b_{0,0}(x)&=1,\\b_{0,1}(x)&=1-1x,&b_{1,1}(x)&=0+1x\\b_{0,2}(x)&=1-2x+1x^{2},&b_{1,2}(x)&=0+2x-2x^{2},&b_{2,2}(x)&=0+0x+1x^{2}\\b_{0,3}(x)&=1-3x+3x^{2}-1x^{3},&b_{1,3}(x)&=0+3x-6x^{2}+3x^{3},&b_{2,3}(x)&=0+0x+3x^{2}-3x^{3},&b_{3,3}(x)&=0+0x+0x^{2}+1x^{3}\end{aligned}}}$

Properties

The Bernstein basis polynomials have the following properties:

• ${\displaystyle b_{\nu ,n}(x)=0}$, if ${\displaystyle \nu <0}$ or ${\displaystyle \nu >n.}$
• ${\displaystyle b_{\nu ,n}(x)\geq 0}$ for ${\displaystyle x\in [0,\ 1].}$
• ${\displaystyle b_{\nu ,n}\left(1-x\right)=b_{n-\nu ,n}(x).}$
• ${\displaystyle b_{\nu ,n}(0)=\delta _{\nu ,0}}$ and ${\displaystyle b_{\nu ,n}(1)=\delta _{\nu ,n}}$ where ${\displaystyle \delta }$ is the Kronecker delta function: ${\displaystyle \delta _{ij}={\begin{cases}0&{\text{if }}i\neq j,\\1&{\text{if }}i=j.\end{cases}}}$
• ${\displaystyle b_{\nu ,n}(x)}$ has a root with multiplicity ${\displaystyle \nu }$ at point ${\displaystyle x=0}$ (note: if ${\displaystyle \nu =0}$, there is no root at 0).
• ${\displaystyle b_{\nu ,n}(x)}$ has a root with multiplicity ${\displaystyle \left(n-\nu \right)}$ at point ${\displaystyle x=1}$ (note: if ${\displaystyle \nu =n}$, there is no root at 1).
• The derivative can be written as a combination of two polynomials of lower degree:
  $${\displaystyle b'_{\nu ,n}(x)=n\left(b_{\nu -1,n-1}(x)-b_{\nu ,n-1}(x)\right).}$$

  
• The k-th derivative at 0:
  $${\displaystyle b_{\nu ,n}^{(k)}(0)={\frac {n!}{(n-k)!}}{\binom {k}{\nu }}(-1)^{\nu +k}.}$$

  
• The k-th derivative at 1:
  $${\displaystyle b_{\nu ,n}^{(k)}(1)=(-1)^{k}b_{n-\nu ,n}^{(k)}(0).}$$

  
• The transformation of the Bernstein polynomial to monomials is
  $${\displaystyle b_{\nu ,n}(x)={\binom {n}{\nu }}\sum _{k=0}^{n-\nu }{\binom {n-\nu }{k}}(-1)^{n-\nu -k}x^{\nu +k}=\sum _{\ell =\nu }^{n}{\binom {n}{\ell }}{\binom {\ell }{\nu }}(-1)^{\ell -\nu }x^{\ell },}$$

  
  and by the inverse binomial transformation, the reverse transformation is^[2]
  $${\displaystyle x^{k}=\sum _{i=0}^{n-k}{\binom {n-k}{i}}{\frac {1}{\binom {n}{i}}}b_{n-i,n}(x)={\frac {1}{\binom {n}{k}}}\sum _{j=k}^{n}{\binom {j}{k}}b_{j,n}(x).}$$

  
• The indefinite integral is given by
  $${\displaystyle \int b_{\nu ,n}(x)\,dx={\frac {1}{n+1}}\sum _{j=\nu +1}^{n+1}b_{j,n+1}(x).}$$

  
• The definite integral is constant for a given n:
  $${\displaystyle \int _{0}^{1}b_{\nu ,n}(x)\,dx={\frac {1}{n+1}}\quad \ \,{\text{for all }}\nu =0,1,\dots ,n.}$$

  
• If ${\displaystyle n\neq 0}$, then ${\displaystyle b_{\nu ,n}(x)}$ has a unique local maximum on the interval ${\displaystyle [0,\,1]}$ at ${\displaystyle x={\frac {\nu }{n}}}$. This maximum takes the value
  $${\displaystyle \nu ^{\nu }n^{-n}\left(n-\nu \right)^{n-\nu }{n \choose \nu }.}$$

  
• The Bernstein basis polynomials of degree ${\displaystyle n}$ form a partition of unity:
  $${\displaystyle \sum _{\nu =0}^{n}b_{\nu ,n}(x)=\sum _{\nu =0}^{n}{n \choose \nu }x^{\nu }\left(1-x\right)^{n-\nu }=\left(x+\left(1-x\right)\right)^{n}=1.}$$

  
• By taking the first ${\displaystyle x}$-derivative of ${\displaystyle (x+y)^{n}}$, treating ${\displaystyle y}$ as constant, then substituting the value ${\displaystyle y=1-x}$, it can be shown that
  $${\displaystyle \sum _{\nu =0}^{n}\nu b_{\nu ,n}(x)=nx.}$$

  
• Similarly the second ${\displaystyle x}$-derivative of ${\displaystyle (x+y)^{n}}$, with ${\displaystyle y}$ again then substituted ${\displaystyle y=1-x}$, shows that
  $${\displaystyle \sum _{\nu =1}^{n}\nu (\nu -1)b_{\nu ,n}(x)=n(n-1)x^{2}.}$$

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

  
• The expansion of the Chebyshev Polynomials of the First Kind into the Bernstein basis is^[3]
  $${\displaystyle T_{n}(u)=(2n-1)!!\sum _{k=0}^{n}{\frac {(-1)^{n-k}}{(2k-1)!!(2n-2k-1)!!}}b_{k,n}(u).}$$

  

Approximating continuous functions

Let ƒ be a continuous function on the interval [0, 1]. Consider the Bernstein polynomial

${\displaystyle B_{n}(f)(x)=\sum _{\nu =0}^{n}f\left({\frac {\nu }{n}}\right)b_{\nu ,n}(x).}$

It can be shown that

${\displaystyle \lim _{n\to \infty }{B_{n}(f)}=f}$

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 ${\displaystyle \mathbb {R} }$.^[7]

A more general statement for a function with continuous k^th derivative is

${\displaystyle {\left\|B_{n}(f)^{(k)}\right\|}_{\infty }\leq {\frac {(n)_{k}}{n^{k}}}\left\|f^{(k)}\right\|_{\infty }\quad \ {\text{and}}\quad \ \left\|f^{(k)}-B_{n}(f)^{(k)}\right\|_{\infty }\to 0,}$

where additionally

${\displaystyle {\frac {(n)_{k}}{n^{k}}}=\left(1-{\frac {0}{n}}\right)\left(1-{\frac {1}{n}}\right)\cdots \left(1-{\frac {k-1}{n}}\right)}$

is an eigenvalue of B_n; the corresponding eigenfunction is a polynomial of degree k.

Probabilistic proof

This proof follows Bernstein's original proof of 1912.^[8] See also Feller (1966) or Koralov & Sinai (2007).^[9][5]

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 ${\displaystyle \operatorname {\mathcal {E}} \left[{\frac {K}{n}}\right]=x\ }$ and

${\displaystyle p(K)={n \choose K}x^{K}\left(1-x\right)^{n-K}=b_{K,n}(x)}$

By the weak law of large numbers of probability theory,

${\displaystyle \lim _{n\to \infty }{P\left(\left|{\frac {K}{n}}-x\right|>\delta \right)}=0}$

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

${\displaystyle \lim _{n\to \infty }{P\left(\left|f\left({\frac {K}{n}}\right)-f\left(x\right)\right|>\varepsilon \right)}=0}$

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

${\displaystyle \lim _{n\to \infty }{\operatorname {\mathcal {E}} \left(\left|f\left({\frac {K}{n}}\right)-f\left(x\right)\right|\right)}=0}$

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

${\displaystyle \operatorname {\mathcal {E}} \left[f\left({\frac {K}{n}}\right)\right]=\sum _{K=0}^{n}f\left({\frac {K}{n}}\right)p(K)=\sum _{K=0}^{n}f\left({\frac {K}{n}}\right)b_{K,n}(x)=B_{n}(f)(x)}$

Elementary proof

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:

1. ${\displaystyle \sum _{k}{n \choose k}x^{k}(1-x)^{n-k}=1}$ ("probability")
2. ${\displaystyle \sum _{k}{k \over n}{n \choose k}x^{k}(1-x)^{n-k}=x}$ ("mean")
3. ${\displaystyle \sum _{k}\left(x-{k \over n}\right)^{2}{n \choose k}x^{k}(1-x)^{n-k}={x(1-x) \over n}.}$ ("variance")

In fact, by the binomial theorem

and this equation can be applied twice to ${\displaystyle t{\frac {d}{dt}}}$. The identities (1), (2), and (3) follow easily using the substitution ${\displaystyle t=x/(1-x)}$.

Within these three identities, use the above basis polynomial notation

${\displaystyle b_{k,n}(x)={n \choose k}x^{k}(1-x)^{n-k},}$

and let

${\displaystyle f_{n}(x)=\sum _{k}f(k/n)\,b_{k,n}(x).}$

Thus, by identity (1)

${\displaystyle f_{n}(x)-f(x)=\sum _{k}[f(k/n)-f(x)]\,b_{k,n}(x),}$

so that

${\displaystyle |f_{n}(x)-f(x)|\leq \sum _{k}|f(k/n)-f(x)|\,b_{k,n}(x).}$

Since f is uniformly continuous, given ${\displaystyle \varepsilon >0}$, there is a ${\displaystyle \delta >0}$ such that ${\displaystyle |f(a)-f(b)|<\varepsilon }$ whenever ${\displaystyle |a-b|<\delta }$. Moreover, by continuity, ${\displaystyle M=\sup |f|<\infty }$. But then

${\displaystyle |f_{n}(x)-f(x)|\leq \sum _{|x-{k \over n}|<\delta }|f(k/n)-f(x)|\,b_{k,n}(x)+\sum _{|x-{k \over n}|\geq \delta }|f(k/n)-f(x)|\,b_{k,n}(x).}$

The first sum is less than ε. On the other hand, by identity (3) above, and since ${\displaystyle |x-k/n|\geq \delta }$, the second sum is bounded by ${\displaystyle 2M}$ times

${\displaystyle \sum _{|x-k/n|\geq \delta }b_{k,n}(x)\leq \sum _{k}\delta ^{-2}\left(x-{k \over n}\right)^{2}b_{k,n}(x)=\delta ^{-2}{x(1-x) \over n}<{1 \over 4}\delta ^{-2}n^{-1}.}$

(Chebyshev's inequality)

It follows that the polynomials f_n tend to f uniformly.

Generalizations to higher dimension

Bernstein polynomials can be generalized to k dimensions – the resulting polynomials have the form B_i1(x₁) B_i2(x₂) ... B_ik(x_k).^[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] × ... × [a_k, b_k]. For a continuous function f on the k-fold product of the unit interval, the proof that f(x₁, x₂, ... , x_k) can be uniformly approximated by

${\displaystyle \sum _{i_{1}}\sum _{i_{2}}\cdots \sum _{i_{k}}{n_{1} \choose i_{1}}{n_{2} \choose i_{2}}\cdots {n_{k} \choose i_{k}}f\left({i_{1} \over n_{1}},{i_{2} \over n_{2}},\dots ,{i_{k} \over n_{k}}\right)x_{1}^{i_{1}}(1-x_{1})^{n_{1}-i_{1}}x_{2}^{i_{2}}(1-x_{2})^{n_{2}-i_{2}}\cdots x_{k}^{i_{k}}(1-x_{k})^{n_{k}-i_{k}}}$

is a straightforward extension of Bernstein's proof in one dimension. ^[14]

See also

• Polynomial interpolation
• Newton form
• Lagrange form
• Binomial QMF (also known as Daubechies wavelet)

Notes

References

• 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
• Lorentz, G. G. (1953), Bernstein Polynomials, University of Toronto Press
• 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
• Burkill, J. C. (1959), Lectures On Approximation By Polynomials (PDF), Bombay: Tata Institute of Fundamental Research, pp. 7–8
• 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.
• Korovkin, P.P. (2001) [1994], "Bernstein polynomials", Encyclopedia of Mathematics, EMS Press
• 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
• Beals, Richard (2004), Analysis. An introduction, Cambridge University Press, pp. 95–98, ISBN 0521600472

External links

• 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.
• Petrone, Sonia (1999). "Random Bernstein polynomials". Scand. J. Stat. 26 (3): 373–393. doi:10.1111/1467-9469.00155. S2CID 122387975.
• Oruc, Halil; Phillips, Geoerge M. (1999). "A generalization of the Bernstein Polynomials". Proceedings of the Edinburgh Mathematical Society. 42 (2): 403–413. doi:10.1017/S0013091500020332.
• 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.
• Casselman, Bill (2008). "From Bézier to Bernstein". Feature Column from American Mathematical Society
• 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.
• Weisstein, Eric W. "Bernstein Polynomial". MathWorld.
• This article incorporates material from properties of Bernstein polynomial on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.