Bernstein's constant

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... .^[1]

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|, Bernstein^[2] showed that the limit

${\displaystyle \beta =\lim _{n\to \infty }2nE_{2n}(f),\,}$

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

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

was disproven by Varga and Carpenter,^[3] who calculated

${\displaystyle \beta =0.280169499023\dots \,.}$