# 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 |