Abstract
We describe the functions needed in the determination of the rate of convergence of best \(L^\infty \) rational approximation to \(\exp ( - x)\) on [0,∞) when the degree n of the approximation tends to ∞ (“1/9” problem).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N.I. Akhiezer, Elements of the Theory of Elliptic Functions, 2nd ed. (Nauka, Moscow, 1970); Translations of Mathematical Monographs, Vol. 79 (Amer. Math. Soc., Providence, RI, 1990).
C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, The PARI-GP Calculator, guides and software at ftp://megrez.math.u-bordeaux.fr/pub/pari/.
J.M. Borwein and P.B. Borwein, Pi and the AGM (Wiley, New York, 1987).
A.J. Carpenter, A. Ruttan and R.S. Varga, Extended numerical computations on the “1=9” conjecture in rational approximation theory, in: Rational Approximation and Interpolation, eds. P.R. Graves-Morris, E.B. Saff and R.S.Varga, Lecture Notes in Mathematics, Vol. 1105 (Springer, Berlin, 1984) pp. 383–411.
E.T. Copson, Partial Differential Equations (Cambridge Univ. Press, Cambridge, 1975).
A. Erdlyi et al., Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).
A.A. Gonchar and E.A. Rakhmanov, Equilibrium distribution and the degree of rational approximation of analytic functions, Mat. Sbornik 134(176) (1987) 306–352; Math. USSR Sb. 62 (1989) 305–348.
G.H. Halphen, Trait des Fonctions Elliptiques et de Leurs Applications. Première Partie. Thorie des Fonctions Elliptiques et de Leurs Développements en Sries (Gauthier-Villars, Paris, 1886). Also available at http://moa.cit.cornell.edu/.
P. Henrici, Applied and Computational Complex Analysis, Vol. 3 (Wiley, New York, 1986).
Jahnke, Emde and Lösch, Tables of Higher Functions-Tafeln höherer Funktionen, 6th ed. (Teubner, Stuttgart, 1960).
K. Knopp, Theory and Application of Infinite Series, 2nd English ed. (Blackie & Son, 1951).
H. Kober, Dictionary of Conformal Representations, 2nd ed. (Dover, New York, 1957).
D.S. Lubinsky, Orthogonal polynomials for exponential weights, Acta Appl. Math.
A.P. Magnus, On the use of Carathodory-Fejr method for investigating “1=9” and similar constants, in: Nonlinear Numerical Methods and Rational Approximation, ed. A. Cuyt (Reidel, Dordrecht, 1988) pp. 105–132.
A.P. Magnus, Asymptotics and super asymptotics of best rational approximation error norms for the exponential function (the “1=9” problem) by the Carathodory-Fejr method, in: Nonlinear Methods and Rational Approximation II, ed. A. Cuyt (Kluwer, Dordrecht, 1994) pp. 173–185.
T.H. Southard, Weierstrass elliptic and related functions, in: Handbook of Mathematical Functions (National Bureau of Standards, 1964; Dover, 1965) chapter 18.
H. Stahl, On the convergence of generalized Pad approximants, Constr. Approx. 5 (1989) 221–240.
H. Stahl, Convergence of rational interpolants, Bull. Belg. Math. Soc.-Simon Stevin Suppl. (1996) 11–32.
L.N. Trefethen and M. Gutknecht, The Carathodory-Fejr method for real rational approximation, SIAM J. Numer. Anal. 20 (1983) 420–436.
R.S. Varga, Scientific Computation on Mathematical Problems and Conjectures, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 60 (SIAM, Philadelphia, PA, 1990).
H. Wallin, Potential theory and approximation of analytic functions by rational interpolation, in: Lecture Notes in Mathematics, Vol. 797 (Springer, Berlin, 1979) pp. 434–450.
J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 4th ed. (Amer. Math. Soc., Providence, RI, 1965).
Rights and permissions
About this article
Cite this article
Magnus, A.P., Meinguet, J. The elliptic functions and integrals of the “\({1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}\)” problem. Numerical Algorithms 24, 117–139 (2000). https://doi.org/10.1023/A:1019141226189
Issue Date:
DOI: https://doi.org/10.1023/A:1019141226189