On the bernstein conjecture in approximation theory
Tóm tắt
WithE
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 constantβ for which lim 2nE
2n
(|x|)=β.n→∞ Moreover, by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds forβ: 0.278<β<0.286. Now, the average of these bounds is 0.282, which, as Bernstein noted as a “curious coincidence,” is very close to 1/(2√π)=0.2820947917... This observation has over the years become known as the Bernstein Conjecture: Isβ=1/(2√π)? We show here that the Bernstein conjecture isfalse. In addition, we determine rigorous upper and lower bounds forβ, and by means of the Richardson extrapolation procedure, estimateβ to approximately 50 decimal places.
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