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.
\Bibitem{VarCar86}
\by R.~S.~Varga, A.~J.~Carpenter
\paper On a~conjecture of S.~Bernstein in approximation theory
\jour Math. USSR-Sb.
\yr 1987
\vol 57
\issue 2
\pages 547--560
\mathnet{http://mi.mathnet.ru/eng/sm1844}
\crossref{https://doi.org/10.1070/SM1987v057n02ABEH003086}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=842399}
\zmath{https://zbmath.org/?q=an:0661.41005}
Linking options:
https://www.mathnet.ru/eng/sm1844
https://doi.org/10.1070/SM1987v057n02ABEH003086
https://www.mathnet.ru/eng/sm/v171/i4/p535
This publication is cited in the following 9 articles:
P. Kubelík, V. G. Kurbatov, I. V. Kurbatova, “Calculating a function of a matrix with a real spectrum”, Numer Algor, 90:3 (2022), 905
Yanjun Han, Jiantao Jiao, Rajarshi Mukherjee, “On estimation of $L_{r}$-norms in Gaussian white noise models”, Probab. Theory Relat. Fields, 177:3-4 (2020), 1243
T. Tony Cai, Mark G. Low, “Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional”, Ann. Statist., 39:2 (2011)
Amos J. Carpenter, “Scientific computation on some mathematical problems”, Journal of Computational and Applied Mathematics, 66:1-2 (1996), 111
Vasilev R., “Asymptotic of the Best Uniform Approximations on Compacts of the Real Axis for Several Lipschitz Type Functions”, Dokl. Akad. Nauk, 339:5 (1994), 588–590
Stahl H., “Best Uniform Rational Approximation of X-Alpha on [0, 1]”, Bull. Amer. Math. Soc., 28:1 (1993), 116–122
H. Stahl, “Best uniform rational approximation of $|x|$ on $[-1,1]$”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 461–487
Varga R., “How High-Precision Calculations Can Stimulate Mathematical Research”, Appl. Numer. Math., 10:3-4 (1992), 177–193
R. S. Varga, A. Ruttan, A. J. Carpenter, “Numerical results on best uniform rational approximation of $|x|$ on $[-1,1]$”, Math. USSR-Sb., 74:2 (1993), 271–290
Citation in format AMSBIB
Contact us: