# Bernstein polynomials

Algebraic polynomials defined by the formula

$$B _ {n} (f; x ) = \ B _ {n} (x ) =$$

$$= \ \sum _ { k=0 } ^ { n } f \left ( { \frac{k}{n} } \right ) \left ( \begin{array}{c} n \\ k \end{array} \right ) x ^ {k} (1-x) ^ {n-k} ,\ n = 1, 2 ,\dots .$$

Introduced by S.N. Bernshtein in 1912 (cf. ). The sequence of Bernstein polynomials converges uniformly to a function $f$ on the segment $0 \leq x \leq 1$ if $f$ is continuous on this segment. For a function which is bounded by $C$, $0 < C < 1$, with a discontinuity of the first kind,

$$B _ {n} (f; C) \rightarrow \ { \frac{f (C _ {-} ) + f (C _ {+} ) }{2} } .$$

The equation

$$B _ {n} (f; c) - f (c) = \ \frac{f ^ { \prime\prime } (c)c(1-c) }{2n} + o \left ( \frac{1}{n} \right )$$

is valid if $f$ is twice differentiable at the point $c$. If the $k$- th derivative $f ^ { (k) }$ of the function is continuous on the segment $0 \leq x \leq 1$, the convergence

$$B _ {n} ^ { (k) } (f; x) \rightarrow f ^ { (k) } (x)$$

is uniform on this segment. A study was made ([1b], ) of the convergence of Bernstein polynomials in the complex plane if $f$ is analytic on the segment $0 \leq x \leq 1$.

How to Cite This Entry:
Bernstein polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein_polynomials&oldid=46028
This article was adapted from an original article by P.P. Korovkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article