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 approximately 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"|, harvtxt|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:

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

was disproven by harvtxt|Varga|Carpenter|1987, who calculated

:

References

*citation|last=Bernstein|first= S. N. |title=Sur la meilleure approximation de |x| par les polynomes de degrés donnés|journal= Acta Math. |volume=37|pages= 1-57|year= 1914 |doi=10.1007/BF02401828
*citation|last=Varga|first= Richard S.|last2= Carpenter|first2= Amos J. |title=A conjecture of S. Bernstein in approximation theory|journal= Math. USSR Sbornik |volume=57|pages= 547-560|year= 1987
id=MR|0842399
DOI = 10.1070/SM1987v057n02ABEH003086
*citation|last=Weisstein|first= Eric W. |chapter=Bernstein's Constant|title= From MathWorld--A Wolfram Web Resource|url= http://mathworld.wolfram.com/BernsteinsConstant.html