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Extended numerical computations on the ”1/9” conjecture in rational approximation theory. (English) Zbl 0553.41022
Rational approximation and interpolation, Proc. Conf., Tampa/Fla. 1983, Lect. Notes Math. 1105, 383-411 (1984).
[For the entire collection see Zbl 0544.00011.]
The behavior of the constants \(\lambda_{n,n}(e^{-x})\), denoting the errors of best uniform approximation to \(e^{-x}\) on the interval \([0,+\infty)\) by real rational functions having numerator and denominator polynomials of degree at most n, has generated much recent interest in the approximation theory literature. Based on high-precision calculations with 230 decimal digits, we present the table of constants \(\{\lambda_{n,n}(e^{-x})\}^{30}_{n=0}\), rounded to forty siginificant digits, and we discuss their significance to related conjectures in this area. In particular, on applying repeated Richardson extrapolation (with \(x_ n=1/n^ 2)\) to the ratios \(\{\lambda_{n-1,n- 1}(e^{-x})/\lambda_{n,n}(e^{-x})\}^{30}_{n=1}\), it appears numerically that \[ \overline{\lim}_{n\to \infty}\lambda^{1/n}_{n,n}(e^{-x})\Doteq 1/9.289\quad 025\quad 491\quad 920\quad 81, \] which is significantly smaller than the ratio 1/9.037 recently derived by Opitz and Scherer.

41A20 Approximation by rational functions
41A25 Rate of convergence, degree of approximation