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On a conjecture of S. Bernstein in approximation theory
R. S. Varga,
A. J. Carpenter
Abstract:
With
$E_{2n}(|x|)$ denoting the error of best uniform approximation to
$|x|$ by polynomials of degree at most
$2n$ on the interval
$[-1,1]$, the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constant
$\beta$ for which
$$
\lim_{n\to\infty}(2nE_{2n}(|x|))=:\beta.
$$
Moreover, by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds for
$\beta$:
$0,278<\beta<0,286$ Now, the average of these bounds is 0.282, which, as Bernstein noted as a “curious coincidence”, is very close to
$\frac1{2\sqrt\pi}=0,2820947917\dots$. This observation has over the years become known as
The Bernstein Conjecture. {\it Is
$\beta=\frac1{2\sqrt\pi}?$}
We show here that the Bernstein conjecture is false. In addition, we determine rigorous upper and lower bounds for
$\beta$, and by means of the Richardson extrapolation procedure, estimate
$\beta$ to approximately 50 decimal places.
Tables: 4.
Bibliography: 12 titles.
UDC:
517.5
MSC: 41A25 Received: 27.03.1985