Amber polynomial

The Bernstein polynomials (after Sergei Natanowitsch Bernstein ) are a special family of real polynomials with integer coefficients.

Use and history

The Bernstein polynomials have their origin in approximation theory . With their help, their discoverer, Bernstein, was able to provide a constructive proof of Weierstrass's approximation theorem in 1911 . At the end of the 1950s there were first attempts to use methods based on Bernstein polynomials in the design of curves and surfaces. Paul de Faget de Casteljau at Citroën and Pierre Bézier at Renault used the amber polynomials in their development of Bézier curves and thus laid the foundation for today's Computer Aided Design (CAD) .

definition

For are the real polynomials ${\displaystyle n\in \mathbb {N} _{0}}$

${\displaystyle B_{i,n}\colon \mathbb {R} \to \mathbb {R} ,\;t\mapsto {n \choose i}\,t^{i}\,(1-t)^{n-i}}$

(with ) the Bernstein polynomials of degree . ${\displaystyle 0\leq i\leq n}$ ${\displaystyle n}$

The generalized Bernstein polynomials are obtained by affine transformation (mapping the interval to any interval ) ${\displaystyle [0,1]}$ ${\displaystyle [a,b]}$

${\displaystyle B_{i,n}^{[a,b]}\colon \mathbb {R} \to \mathbb {R} ,\;t\mapsto {\frac {1}{(b-a)^{n}}}{n \choose i}(t-a)^{i}\,(b-t)^{n-i}}$.

Here designated

${\displaystyle {n \choose i}={\frac {n!}{i!(n-i)!}}}$

the binomial coefficient .

example

The following figure shows the Bernstein polynomials , of degree : ${\displaystyle B_{i,4}}$ ${\displaystyle 0\leq i\leq 4}$ ${\displaystyle 4}$

properties

The Bernstein polynomials with respect to the interval have the following properties: ${\displaystyle [0,1]}$

• Basic property : The Bernstein polynomials are linearly independent and form a basis of , the space of the polynomials of degree less than or equal . ${\displaystyle \{B_{i,n}:0\leq i\leq n\}}$ ${\displaystyle \Pi _{n}}$ ${\displaystyle n}$
• Positivity :

  ${\displaystyle B_{i,n}(t)>0}$for everyone . ${\displaystyle t\in (0,1)}$

• Extremes : has exactly one (absolute) maximum in the interval . It's in the place . In particular one obtains: ${\displaystyle B_{i,n}}$ ${\displaystyle [0,1]}$ ${\displaystyle t={\frac {i}{n}}}$

  ${\displaystyle B_{0,n}(0)=B_{n,n}(1)=1}$

• Decomposition of one (also partition of one) :

  ${\displaystyle \sum _{i=0}^{n}B_{i,n}(t)=\sum _{i=0}^{n}{n \choose i}t^{i}(1-t)^{n-i}=1}$

(Result by means of the binomial theorem in .) ${\displaystyle (t+(1-t))^{n}}$

• Symmetry :

  ${\displaystyle B_{i,n}(t)=B_{n-i,n}(1-t)}$

• Recursion formula :

  ${\displaystyle B_{i,n}(t)=(1-t)\cdot B_{i,n-1}(t)+t\cdot B_{i-1,n-1}(t)}$, with the definition

  ${\displaystyle B_{i,n}:=0}$for or ${\displaystyle i<0}$ ${\displaystyle i>n}$

  ${\displaystyle B_{0,0}:=1}$

• Elevation :

  ${\displaystyle B_{i,n}(t)={\frac {i+1}{n+1}}\cdot B_{i+1,n+1}(t)+{\frac {n+1-i}{n+1}}\cdot B_{i,n+1}(t)}$

• Derivatives :

  ${\displaystyle B'_{i,n}(t)=n\left[B_{i-1,n-1}(t)-B_{i,n-1}(t)\right]}$, with the definition

  ${\displaystyle B_{-1,n-1}=B_{n,n-1}:=0}$

• Antiderivative :

  ${\displaystyle \int B_{i,n}\!\left(t\right)dt={\frac {1}{n+1}}\sum \limits _{k=i+1}^{n+1}B_{k,n+1}\!\left(t\right)}$

Approximation by Bernstein polynomials

For a function that means through ${\displaystyle f\colon [0,1]\to \mathbb {R} }$

${\displaystyle B_{n}(f)(t)=\sum _{i=0}^{n}B_{i,n}(t)\cdot f\left({\frac {i}{n}}\right)}$

Polynomial defined the -th Bernstein polynomial of the function . ${\displaystyle B_{n}(f)}$ ${\displaystyle n}$ ${\displaystyle f}$

If there is a continuous function on the interval , the sequence of its Bernstein polynomials converges uniformly to . ${\displaystyle f}$ ${\displaystyle [0,1]}$ ${\displaystyle B_{n}(f)}$ ${\displaystyle f}$

The proof of this theorem can be carried out with the help of the weak law of large numbers or Korovkin's theorem .

Web links

• Bernstein Polynomials (Applet)

literature

• Bernstein, SN, Demonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Soc. Math. Kharkov, Vol. 12, No. 2, pp. 1-2, 1912/1913.