Bernsteinpolynomi

Bernstein polynomials are a specific type of polynomial that form the basis of Bernstein's theorem. They are defined over an interval, typically [0, 1]. For a non-negative integer n, the Bernstein basis polynomials of degree n are given by the formula B_{i,n}(x) = \binom{n}{i} x^i (1-x)^{n-i} for i = 0, 1, ..., n. Here, \binom{n}{i} represents the binomial coefficient, which is the number of ways to choose i items from a set of n items.

Any polynomial p(x) of degree n can be expressed as a linear combination of these basis polynomials. This representation is called the Bernstein form of the polynomial: p(x) = \sum_{i=0}^n c_i B_{i,n}(x), where c_i are the coefficients. A key property of Bernstein polynomials is that if the coefficients c_i are all non-negative, then the polynomial p(x) is also non-negative over the interval [0, 1].

Bernstein polynomials are fundamental in the theory of approximation. Bernstein's theorem states that for any continuous function f defined on the interval [0, 1], the sequence of Bernstein polynomials corresponding to f converges uniformly to f on [0, 1] as the degree n approaches infinity. This means that any continuous function can be arbitrarily well approximated by Bernstein polynomials.

These polynomials have applications in computer graphics, particularly in the definition of Bézier curves. Bézier curves are defined using a set of control points, and their shape is determined by Bernstein polynomials. They are also used in numerical analysis and approximation theory due to their nice convergence properties.