Bernstein's constant

Template:Short description Template:Use shortened footnotes Bernstein's constant, usually denoted by the Greek letter β (beta), is a mathematical constant named after Sergei Natanovich Bernstein and is equal to 0.2801694990... .Template:R

Definition

Let E_n(ƒ) be the error of the best uniform approximation to a real function ƒ(x) on the interval [−1, 1] by real polynomials of no more than degree n. In the case of ƒ(x) = |x|, BernsteinTemplate:R showed that the limit

$\beta =\underset{n\to \mathrm{\infty }}{\lim }2n{E}_{2n}(f),$

called Bernstein's constant, exists and is between 0.278 and 0.286. His conjecture that the limit is:

$\frac{1}{2\sqrt{\pi }}=0.28209\dots .$

was disproven by Varga and Carpenter,Template:R who calculated

$\beta =0.280169499023\dots .$