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Bernstein's constant

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Mathematical 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 En(ƒ) be the error of the best uniform approximation to a real function ƒ(x) on the interval 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

  1. (sequence A073001 in the OEIS)
  2. Bernstein, S.N. (1914). "Sur la meilleure approximation de x par des polynomes de degrés donnés". Acta Math. 37: 1–57. doi:10.1007/BF02401828.
  3. Varga, Richard S.; Carpenter, Amos J. (1987). "A conjecture of S. Bernstein in approximation theory". Math. USSR Sbornik. 57 (2): 547–560. Bibcode:1987SbMat..57..547V. doi:10.1070/SM1987v057n02ABEH003086. MR 0842399.

Further reading

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