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 $[0,1]$.

$$B_i^n(t)\ge 0\mathit{\hspace{1em}\hspace{1em}}0\le t\le 1$$

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
Canonical name	PropertiesOfBernsteinPolynomial
Date of creation	2013-03-22 17:24:13
Last modified on	2013-03-22 17:24:13
Owner	stitch (17269)
Last modified by	stitch (17269)
Numerical id	10
Author	stitch (17269)
Entry type	Derivation
Classification	msc 65D17