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

DefinitionEdit

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 (1914) showed that the limit

 

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

 

was disproven by Varga & Carpenter (1987), who calculated

 

ReferencesEdit

  • 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.