Difference between revisions of "Bernstein polynomial"

Latest revision as of 10:44, 25 November 2016

Definition

The $n$^th Bernstein polynomial for a function $f:I\rightarrow\mathbb{R}$ (where $I:\eq[0,1]\subset\mathbb{R}$ is the closed unit interval) is defined as follows:

• $$\mathcal{B}_n(f;x):\eq\sum^n_{i\eq 0}{}^nC_if\left(\tfrac{k}{n}\right)x^i(1-x)^{n-i}$$

Comments

Notice that the binomial expansion of $(x+(1-x))^n\eq 1^n\eq 1$ is $\sum^n_{i\eq 0}{}^nC_ix^i(1-x)^{n-i}$ so we see: $\sum^n_{i\eq 0}{}^nC_ix^i(1-x)^{n-i}\eq 1$ provided $\vert x\vert < 1$ and $\vert 1-x\vert < 1$, we also note that if $x\eq 0$ or $x\eq 1$ then it is also equal to 1

References