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... .
Definition Let En(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 showed that the limit
β = lim n → ∞ 2 n E 2 n ( f ) , {\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:
1 2 π = 0.28209 … . {\displaystyle {\frac {1}{2{\sqrt {\pi }}}}=0.28209\dots \,.}
was disproven by Varga and Carpenter, who calculated
β = 0.280169499023 … . {\displaystyle \beta =0.280169499023\dots \,.}
References
Further reading Weisstein, Eric W. "Bernstein's Constant". MathWorld.