Rational Approximation to e-x and Related L2-Problems - Schönhage, Arnold (SIAM journal on numerical analysis, 1982)
Rational Chebyshev approximation to e-x on [0, ∞) is known to yield geometric convergence. For the restricted case of rational approximants 1/hn the corresponding limit ratio is 1/3. In case of general rational functions g/h only upper and lower bounds are known; the precise value has been conjectured to be σ = 1/9. The main subject of this paper is the analysis and solution of two related problems in the Hilbert space L2 of the Laguerre polynomials. Problem (A) indeed comes out with the limit 1/9 when applied to distinguished values of the parameters involved, but this has no immediate impact on the original problem of rational approximation. Problem (B) stems from a different mode of normalizing such that it can be used for deriving new lower bounds. In this way $σ \geqq (2 - \sqrt 3)^2$ is obtained.