On the Bernstein conjecture in approximation theory
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 1/(2\(\sqrt{\pi})=0.2820947917..\). This observation has over the years become known as the Bernstein Conjecture: Is \(\beta =1/(2\sqrt{\pi})?\) 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.
scientific article; zbMATH DE number 4058128