Bernstein polynomials

A constructive proof of this theorem (for ƒ real-valued) using Bernstein polynomials is outlined on that page.

Sometimes it is desirable to express the Bézier curve as a polynomial instead of a sum of less straightforward Bernstein polynomials.

The "Bernstein" column shows the decomposition of the Hermite basis functions into Bernstein polynomials of order 3:

With the advent of computer graphics, Bernstein polynomials, restricted to the interval x [0, 1], became important in the form of Bézier curves.

It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related to the Bernstein polynomials used in approximation theory.

The numerator is a weighted Bernstein-form Bézier curve and the denominator is a weighted sum of Bernstein polynomials.

The Bernstein polynomials of a fixed degree m are a family of m+1 linearly independent polynomials that are a partition of unity for the unit interval .

In functional analysis, a branch of mathematics, the Baskakov operators are generalizations of Bernstein polynomials, Szász-Mirakyan operators, and Lupas operators.

Bernstein polynomials thus afford one way to prove the Stone-Weierstrass theorem that every real-valued continuous function on a real interval 'a','b' can be uniformly approximated by polynomial functions over 'R'.

In functional analysis, a discipline within mathematics, the Szász-Mirakyan operators (also spelled "Mirakjan" and "Mirakian") are generalizations of Bernstein polynomials to infinite intervals, introduced by Otto Szász in 1950 and G. M. Mirakjan in 1941.