BernsteinTheorem

Bernstein's theorem, also known as the Bernstein polynomial approximation theorem, is a foundational result in approximation theory. It states that every continuous function on the closed interval [0,1] can be uniformly approximated by Bernstein polynomials. The Bernstein operator B_n acts on a function f by the formula B_n(f)(x) = sum_{k=0}^n f(k/n) binom(n,k) x^k (1−x)^{n−k}, for x in [0,1]. As n increases, B_n(f) converges uniformly to f on [0,1]. This provides a constructive proof of the Weierstrass approximation theorem, giving explicit polynomial approximants rather than an abstract existence statement.

Key properties include that the operator preserves constants and reproduces affine (linear) functions exactly: for f(x) = 1 and f(x) = x, B_n(f) = f for all n. The convergence rate is governed by the smoothness of f and its modulus of continuity; in general, |B_n(f)(x) − f(x)| can be bounded in terms of ω_f, the modulus of continuity of f. A probabilistic interpretation describes B_n(f)(x) as the expectation E[f(K/n)] where K is binomially distributed with parameters n and x.

Extensions and generalizations include multivariate versions on [0,1]^d, Bernstein operators on other domains, and various refinements used in approximation theory. Bernstein polynomials have influenced numerical analysis, wavelet theory, and computer graphics as simple, constructive tools for polynomial approximation. The theorem is named for Sergei N. Bernstein, who introduced it in 1912.