Bernstein's constant

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Binary 0.01000111101110010011000000110011…
Decimal 0.280169499…
Hexadecimal 0.47B930338AAD…
Continued fraction \cfrac{1}{3 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{3 + \cfrac{1}{9+ \ddots}}}}}

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[edit]

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 (1914) 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 & Carpenter (1987), who calculated

\beta=0.280169499023\dots\,.

References[edit]