Bernstein polynomial

A Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials.

Definition

The $n+1$ Bernstein basis polynomials of degree $n$ are defined as

$b_{\nu ,n}(x)=\left(\genfrac{}{}{0ex}{}{n}{\nu }\right)x^{\nu }(1-x{)}^{n-\nu },\text{where }\nu =0,\cdots ,n$.

The name is from the fact that the Bernstein basis polynomials of degree $n$ form a basis for the vector space ${\mathbb{P}}_n$, a space which consists of polynomials with degree at most $n$.

A Bernstein polynomial is a linear combination of Bernstein basis polynomials:

$B_n(x)=\sum _{\nu =0}^n{\beta }_{\nu }{b}_{\nu ,n}(x)$

where the coefficients ${\beta }_{\nu }$ are called Bernstein coefficients.

Examples

Approximation of continuous functions

Let $f$ be a continuous funciton on the interval $[0,1]$. Let ${\beta }_{\nu }=f\left(\frac{\nu }{n}\right)$. We can denote the corresponding Bernstein polynomial as $B_n(x;f)$. The corresponding Bernstein polynomial becomes:

$$\begin{array}{rl}{B}_n(x;f)& =\sum _{\nu =0}^{n}f\left(\frac{\nu }{n}\right)b_{\nu ,n}(x)\\ & =\sum _{\nu =0}^{n}f\left(\frac{\nu }{n}\right)\left(\genfrac{}{}{0ex}{}{n}{\nu }\right)x^{\nu }(1-x{)}^{n-\nu }.\end{array}$$

Properties

Property #1

$B_n(x;1)=1$

Property #2

$B_n(x;x)=x$

Property #3

$B_n(x;x^2)=x^2+\frac{1}{n}(x-x^2)$

Property #4

$B_n(x;af+g)=a{B}_n(x;f)+B_n(x;g)$