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

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)

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

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

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

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)