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... (sequence A073001 in the OEIS).

Binary	0.01000111101110010011000000110011…
Decimal	0.280169499…
Hexadecimal	0.47B930338AAD…
Continued fraction	${\displaystyle {\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{9+\ddots }}}}}}}}}}}$

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 (1914) 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 & Carpenter (1987), who calculated

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

References

• Bernstein, S. N. (1914), "Sur la meilleure approximation de x par des polynomes de degrés donnés" (PDF), Acta Math., 37: 1–57, doi:10.1007/BF02401828
• Varga, Richard S.; Carpenter, Amos J. (1987), "A conjecture of S. Bernstein in approximation theory", Math. USSR Sbornik, 57 (2): 547–560, doi:10.1070/SM1987v057n02ABEH003086, MR 0842399
• Weisstein, Eric W. "Bernstein's Constant". MathWorld.