ON A CONJECTURE OF S. BERNSTEIN IN APPROXIMATION THEORY
Richard S. Varga, A D Karpenter · Mathematics of the USSR-Sbornik · 1987
With denoting the error of best uniform approximation to by polynomials of degree at most on the interval , the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constant for which Moreover, by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds for : Now, the average of these bounds is 0.282, which, as Bernstein noted as a curious coincidence, is very close to . This observation has over the years become known as the Bernstein Conjecture: Is ? We show here that the Bernstein conjecture is false. In addition, we determine rigorous upper and lower bounds for , and by means of the Richardson extrapolation procedure, estimate to approximately 50 decimal places.Tables: 4. Figures: 1. Bibliography: 12 titles.