Bernsteinpolynomokban

Bernstein polynomial bases are a set of polynomials closely related to the construction of Bézier curves. For a given non-negative integer n, the n-th degree Bernstein basis polynomials are defined as follows: 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. These polynomials form a basis for the vector space of all polynomials of degree at most n.

A Bernstein polynomial of degree n is a linear combination of these basis polynomials: 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 they are non-negative on the interval [0, 1]. Their convex hull property is also significant: the graph of a Bernstein polynomial lies within the convex hull of its control points (c_0, c_1, ..., c_n).

Bernstein polynomials are particularly useful in computer-aided geometric design (CAGD) and approximation theory. They provide a stable and intuitive way to define curves and surfaces. The coefficients c_i can be interpreted as control points that influence the shape of the resulting curve. As n increases, the Bernstein polynomial approximation of a continuous function on [0, 1] converges uniformly to the function. This convergence property makes them valuable for approximating arbitrary functions.