Bernstein's constant
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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... .[1]
Definition
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[2] showed that the limit
- [math]\displaystyle{ \beta=\lim_{n \to \infty}2nE_{2n}(f),\, }[/math]
called Bernstein's constant, exists and is between 0.278 and 0.286. His conjecture that the limit is:
- [math]\displaystyle{ \frac {1}{2\sqrt {\pi}}=0.28209\dots\,. }[/math]
was disproven by Varga and Carpenter,[3] who calculated
- [math]\displaystyle{ \beta=0.280169499023\dots\,. }[/math]
References
- ↑ (sequence A073001 in the OEIS)
- ↑ 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. https://zenodo.org/record/1627064.
- ↑ 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. Bibcode: 1987SbMat..57..547V.
Further reading
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