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... .

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 showed that the limit

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

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

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

was disproven by Varga and Carpenter, who calculated

$\beta =0.280169499023\dots \,.$

Further reading

• Weisstein, Eric W. "Bernstein's Constant". MathWorld.