# properties of Bernstein polynomial

The Bernstein polynomials $B_{i}^{n}(t)$ have the following properties:

## 0.1 Non negativity

The polynomials are non-negative over the interval $[0,1]$.

 $B_{i}^{n}(t)\geq 0\quad\quad 0\leq t\leq 1$

## 0.2 Symmetry

The set of polynomials of degree $n$ is symmetric with respect to $t=1/2$.

 $B_{i}^{n}(t)=B_{n-i}^{n}(1-t)$

## 0.3 Maximum

Each polynomial has only one maximum over the interval $[0,1]$ at $t=\frac{i}{n}$.

## 0.4 Normalization

The set of polynomials of degree $n$ forms a partition of unity.

 $\sum_{i=0}^{n}B_{i}^{n}(t)=1$

## 0.5 Degree raising

A polynomial can always be written as a linear combination of polynomials of higher degree.

 $B_{i}^{n-1}(t)=\frac{n-i}{n}B_{i}^{n}(t)+\frac{i+1}{n}B_{i+1}^{n}(t)$
Title properties of Bernstein polynomial PropertiesOfBernsteinPolynomial 2013-03-22 17:24:13 2013-03-22 17:24:13 stitch (17269) stitch (17269) 10 stitch (17269) Derivation msc 65D17