Abstract
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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R. A. Bell, S. M. Shah (1969):Oscillating polynomials and approximations to |x|. Publ. of the Ramanujan Inst.1:167–177.
S. Bernstein (1914):Sur la meilleure approximation de |x| par des polynomes de degrés donnés. Acta. Math.,37:1–57.
R. Bojanic, J. M. Elkins (1975):Bernstein's constant and best approximation on [0, ∞). Publ. de l'Inst. Math., Nouvelle série,18 (32):19–30.
Richard P. Brent (1978):A FORTRAN multiple-precision arithmetic package. Assoc. Comput. Mach. Trans. Math. Software,4:57–70.
C. Brezinski (1978): Algorithmes d'Accélération de la Convergence. Paris: Éditions Technip.
W. J. Cody, A. J. Strecok, H. C. Thacher, Jr. (1973):Chebyshev approximations for the psi function. Math. of Comp.,27:123–127.
P. Henrici (1974): Applied and Computational Complex Analysis, vol. 1. New York: John Wiley & Sons.
G. Meinardus (1967): Approximations of Functions: Theory and Numerical Methods. New York: Springer-Verlag.
E. Ya. Remez (1934):Sur le calcul effectiv des polynômes d'approximation de Tchebichef. C.R. Acad. Sci. Paris,199:337–340.
T. J. Rivlin (1969): An Introduction to the Approximation of Functions. Waltham, Massachusetts: Blaisdell Publishing Co.
D. A. Salvati (1980): Numerical Computation of Polynomials of Best Uniform Approximation to the Function |x|. Master's Thesis, Ohio State University, 39 pp. Columbus, Ohio.
E. T. Whittaker, G. N. Watson (1962): A Course of Modern Analysis, 4th ed. Cambridge: Cambridge University Press.
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Communicated by P. Henrici.
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Varga, R.S., Carpenter, A.J. On the bernstein conjecture in approximation theory. Constr. Approx 1, 333–348 (1985). https://doi.org/10.1007/BF01890040
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DOI: https://doi.org/10.1007/BF01890040