Geodesic
Numerical results on best uniform rational approximation of $|x|$ on $[-1,1]$
Sbornik. Mathematics, Tome 74 (1993) no. 2, pp. 271-290 Cet article a éte moissonné depuis la source Math-Net.Ru

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With $E_{n,n}(|x|;[-1,1])$ denoting the error of best uniform rational approximation from $\pi_{n,n}$ to $|x|$ on $[-1,1]$, we determine the numbers $\{E_{2n,2n}(|x|;[-1,1])\}_{n=1}^{40}$, where each of these numbers was calculated with a precision of at least 200 significant digits. With these numbers, the Richardson extrapolation method was applied to the products $\{e^{\pi\sqrt{2n}}E_{2n,2n}(|x|;[-1,1])\}_{n=1}^{40}$, and it appears, to at least 10 significant digits, that $$ 8\stackrel{?}{=}\lim_{n\to\infty}e^{\pi\sqrt{2n}}E_{2n,2n}(|x|;[-1,1]), $$ which gives rise to an interesting new conjecture in the theory of rational approximation. 
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R. S. Varga; A. Ruttan; A. J. Carpenter. Numerical results on best uniform rational approximation of $|x|$ on $[-1,1]$. Sbornik. Mathematics, Tome 74 (1993) no. 2, pp. 271-290. http://geodesic.mathdoc.fr/item/SM_1993_74_2_a0/

[1] Bernstein S., “Sur meilleure approximation de $|x|$ par des polynomes de degres donnes”, Acta Math., 37 (1937), 1–57 | DOI

[2] Blatt H.-P., Iserles A. and Saff E. B., “Remarks on the behavior of zeros and poles of best approximating polynomials and rational functions”, Algorithms for Approximation, Institute of Mathematics and Its Application Conference Series, 10, eds. J. C. Mason, M. G. Cox, Claredon Press, Oxford, 1987 | MR

[3] Blatt H.-P., Saff E. B., Totik V., “The distribution o extreme points in best complex polynomial approximation”, Journal Gonstr. Approx., 5 (1989), 357–370 | DOI | MR | Zbl

[4] Borwein P. B., Kroo A., Grothmann B., Saff E. B., “The density of alternation points in rational approximation”, Proc. Amer. Math. Soc., 105 (1989), 881–888 | DOI | MR | Zbl

[5] Brent Richard P., “A FORTRAN multiple-precision arithmetic package”, Assoc. Comput. Math. Trans. Math. Software, 4 (1978), 57–70 | DOI

[6] Brezinski C., Algorithms d'Acceleration de la Convergence, Editions Technip., Paris, 1978 | Zbl

[7] Bulanov A. P., “Asimptotika nailuchshikh priblizhenii $|x|$ ratsionalnymi funktsiyami”, Matem. sb., 76(118) (1968), 288–303 | MR | Zbl

[8] Ganelius T., “Rational approximation to $x^\alpha$ on $[0, 1]$”, Analysis Mathematica, 5 (1979), 19–33 | DOI | MR | Zbl

[9] Gantmakher F. R., Teoriya matrits, Nauka, M., 1967 | MR

[10] Gonchar A. A., “Otsenki rosta ratsionalnykh funktsii i nekotorye ikh prilozheniya”, Matem. sb., 72 (114) (1967), 489–503 | Zbl

[11] Gonchar A. A., Rakhmanov E. A., “Ravnovesnoe raspredelenie i skorost ratsionalnoi approksimatsii analiticheskikh funktsii”, Matem. sb., 134(176) (1987), 306–352 | MR | Zbl

[12] Kadec M. I., “On the distribution of points of maximum deviation in the approximation of continuos functions by polynomials”, Trans. Amer. Math. Soc., 26 (1963), 231–234 | MR

[13] Loeb H. L., “Approximation by generalized rationale”, SIAM J. on Numer. Anal., 3 (1966), 34–55 | DOI | MR | Zbl

[14] Meinardus G., Approximation of Functions: Theory and Numerical Methods, Springer - Verlag Inc., New York, 1967 | MR | Zbl

[15] Newman D. J., “Rational approximation to $|x|$”, Michigan Math. J., 11 (1964), 11–14 | DOI | MR | Zbl

[16] Petrushev P. P., Popov V. A., Rational Approximation of Real Functions, Encyclopedia of Mathematics and Its Applications, 28, Cambridge University Press, Cambridge, 1987 | MR | Zbl

[17] Rivlin T. J., An Introduction ot the Approximation of Functions, Blaisdell Publishing Co, Walthan, Mass, 1969 | Zbl

[18] Szusz P., Turan P., “On the constructive theory of functions, III”, Studia Sci. Math. Hungar, 1 (1966), 315–322 | MR

[19] Turan P., “On the approximation of piecewise analytic functions by rational functions”, Sovremennye problemy teorii analiticheskikh funktsii, Nauka, M., 1966, 296–300 | MR

[20] Varga Richard S., Topics in Polynomial and Rational Interpolation and Approximation, University of Montreal Press, 1982 | MR

[21] Varga Richard S., Sientific Computation on Mathematical Problems and Conjectures, CBMS-NSF Regional Series in Applied Mathematics, 60, 1990 | MR | Zbl

[22] Varga R. S., Karpenter A. Dzh., “Ob odnoi gipoteze Bernshteina v teorii priblizhenii”, Matem. sb., 129 (171) (1986), 535–548 | MR

[23] Vyacheslavov N. S., “O ravnomernom priblizhenii $|x|$ ratsionalnymi funktsiyami”, DAN SSSR, 220:3 (1975), 512–515 | Zbl