Bernstein's constant, usually denoted by the Greek letter β (

beta Beta (, ; uppercase , lowercase , or cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labi ...

), is a

mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...

named after

Sergei Natanovich Bernstein Sergei Natanovich Bernstein (russian: Серге́й Ната́нович Бернште́йн, sometimes Romanized as ; 5 March 1880 – 26 October 1968) was a Ukrainian and Russian mathematician of Jewish origin known for contributions to parti ...

and is equal to 0.2801694990... .

Definition

Let ''E''_''n''(ƒ) be the error of the best uniform approximation to a

real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...

''ƒ''(''x'') on the interval minus;1, 1by real polynomials of no more than degree ''n''. In the case of ''ƒ''(''x'') = , ''x'', , Bernstein showed that the limit :

\beta=\lim_2nE_(f),\,

called Bernstein's constant, exists and is between 0.278 and 0.286. His

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

that the limit is: :

\frac =0.28209\dots\,.

was disproven by Varga and Carpenter, who calculated :

\beta=0.280169499023\dots\,.

Definition

References

Further reading