Editing Bernstein polynomial (section)

==Properties==
The Bernstein basis polynomials have the following properties:
* <math>\ b_{\nu, n}\!(x) \equiv 0\ ,</math> if <math>\ \nu < 0\ </math> or if <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_{i,j}\ </math> is the [[Kronecker delta]] function: <math>\ \delta_{ij} \equiv \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: when <math>\ \nu = 0\ ,</math> there is no root at {{math|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 {{math|1}}).
* The [[derivative]] can be written as a combination of two polynomials of lower degree: <math display="block">\ b_{\nu, n}'\!(x) = n \bigl[\ b_{\nu - 1, n - 1}\!(x)\ -\ b_{\nu, n - 1}\!(x)\ \bigr] ~.</math>
* The {{mvar|k}}-th derivative at {{math|0}}: <math display="block">\ b_{\nu, n}^{(k)}\!(0)\ =\ \frac{n!}{(n - k)!} \binom{k}{\nu} (-1)^{\nu + k} ~.</math>
* The {{mvar|k}}-th derivative at 1: <math display="block">\ b_{\nu, n}^{(k)}(1)\ =\ (-1)^k b_{n - \nu, n}^{(k)}(0) ~.</math>
* The transformation of the Bernstein polynomial to monomials is <math display="block">\ 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 |first=R.J. |last=Mathar |year=2018 |title=Orthogonal basis function over the unit circle with the minimax property |at=Appendix&nbsp;B |eprint=1802.09518 |class=math.NA }}</ref> <math display="block">\ x^k\ =\ \sum_{i=0}^{n-k} \frac{ \binom{n-k}{i} }{ \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 display="block">\ \int b_{\nu, n}\!(x)\ \operatorname{d} x = \frac{1}{n+1} \sum_{j=\nu+1}^{n+1} b_{j, n+1}\!(x) ~.</math> 
* The definite integral is constant for a given {{mvar|n}}: <math display="block">\ \int_0^1 b_{\nu, n}\!(x)\ \operatorname{d} x = \frac{1}{n+1} ~~</math> for all <math>~~ \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 display="block">\ \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 display="block">\ \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 <math>x</math>-derivative of <math>\ (x + y)^n\ ,</math> treating  <math>\ y\ </math> as constant, then substituting the value <math>\ y = 1-x\ ,</math> it can be shown that <math display="block">\ \sum_{\nu=0}^{n} \nu\ b_{\nu, n}\!(x) = n\ x ~.</math>
* Similarly the second <math>\ x\ </math>-derivative of <math>\ (x+y)^n\ ,</math> with <math>\ y\ </math> then again substituted <math>\ y = 1-x\ ,</math> shows that <math display="block">\ \sum_{\nu=1}^{n} \nu \left( \nu-1 \right)\ b_{\nu, n}\!(x) = n\left( n-1 \right)\ x^2 ~.</math>
* A Bernstein polynomial can always be written as a linear combination of polynomials of higher degree: <math display="block">\ b_{\nu, n - 1}\!(x)\ =\ \left( \frac{n - \nu}{n}\right)\ b_{\nu, n}\!(x)\ +\ \left( \frac{\nu + 1}{n}\right)\ b_{\nu + 1, n}\!(x) ~.</math>
* The expansion of the [[Chebyshev polynomials|Chebyshev Polynomials of the First Kind]] into the Bernstein basis is<ref>{{cite journal |first1=Abedallah |last1=Rababah |year=2003 |title=Transformation of Chebyshev-Bernstein polynomial basis |journal=Computational Methods in Applied Mathematics |volume=3 |number=4 |pages=608–622 |s2cid=120938358 |doi=10.2478/cmam-2003-0038 |doi-access=free}}</ref> <math display="block">\ T_n\!(u)\ =\ (2n-1)!!\ \sum_{k=0}^n \frac{~ (-1)^{n-k}\ }{\ (2k-1)!!\ (2n-2k-1)!!\ }\ b_{k,n}\!(u) ~.</math>