On a~conjecture of S.~Bernstein in approximation theory

R. S. Varga ; A. J. Carpenter

Sbornik. Mathematics, Tome 57 (1987) no. 2, pp. 547-560

Résumé

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 $\beta$: $0,278\beta0,286$ 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.