# Bernstein polynomial and Bernstein function

• Tatianaoo
In summary, a Bernstein polynomial is a mathematical function used for approximating other functions by expressing them as a linear combination of Bernstein basis polynomials. These polynomials are always positive and sum to 1 on the interval [0,1]. A Bernstein function is a special type of function that is non-negative, convex, and has a derivative that is also a Bernstein polynomial. They have various applications in mathematics, computer science, and engineering, including curve fitting, numerical integration, and solving differential equations. They are also used in the analysis of algorithms and in probability theory. Bernstein polynomials are the basis functions for Bezier curves, which are commonly used in computer graphics and animation. Additionally, Bernstein polynomials and functions are closely related to the bin

#### Tatianaoo

Hi! Does anybody know if there is something in common between Bernstein functions and Bernstein polynomials except the word 'Bernstein'? I mean from mathematical point of view.

Thanks!

## What is a Bernstein polynomial?

A Bernstein polynomial is a mathematical function that is used to approximate other functions by expressing them as a linear combination of Bernstein basis polynomials. These polynomials are defined on the interval [0,1] and have the property that they are always positive and sum to 1.

## What is a Bernstein function?

A Bernstein function is a special type of function that is defined on the interval [0,1] and has the properties of being non-negative, convex, and has a derivative that is a Bernstein polynomial. These functions are commonly used in statistics and probability theory.

## What are the applications of Bernstein polynomials and functions?

Bernstein polynomials and functions have a wide range of applications in mathematics, computer science, and engineering. They are commonly used for curve fitting, numerical integration, and solving differential equations. They are also used in the analysis of algorithms and in probability theory.

## What is the relationship between Bernstein polynomials and Bezier curves?

Bernstein polynomials are the basis functions used in Bezier curves, which are commonly used in computer graphics and animation. Bezier curves are defined by a series of control points and the Bernstein polynomials are used to calculate the position of points along the curve.

## How are Bernstein polynomials and functions related to the binomial distribution?

The binomial distribution is a probability distribution that is used to model the number of successes in a sequence of independent Bernoulli trials. This distribution is closely related to Bernstein polynomials and functions, as the coefficients of the polynomials correspond to the probabilities of success in each trial.