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... .{{r|OEIS_A073001}}

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{{r|Bernstein1914}} 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,{{r|VargaCarpenter1987}} who calculated

: $\beta=0.280169499023\dots\,.$

References

{{reflist|refs=

{{cite journal |last=Bernstein |first= S.N. |date=1914 |title=Sur la meilleure approximation de x par des polynomes de degrés donnés |journal=Acta Math. |volume=37 |pages=1–57 |doi=10.1007/BF02401828 |doi-access=free |url=https://zenodo.org/record/1627064 }}

{{cite journal |last1=Varga |first1=Richard S. |last2=Carpenter |first2=Amos J. |date=1987 |title=A conjecture of S. Bernstein in approximation theory |journal=Math. USSR Sbornik |volume=57 |issue=2 |pages=547–560 |doi=10.1070/SM1987v057n02ABEH003086 |bibcode=1987SbMat..57..547V |mr=0842399 }}

}}

Bernstein's constant

Definition

References

Further reading