Rational Approximation to e-x and Related L2-Problems

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 conjectur...

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Bibliographic Details
Published inSIAM journal on numerical analysis Vol. 19; no. 5; pp. 1067 - 1080
Main Author Schönhage, Arnold
Format Journal Article
LanguageEnglish
Published Society for Industrial and Applied Mathematics 01.10.1982
Subjects
Approximation
Coefficients
Degrees of polynomials
Geometric angles
Hilbert spaces
Inner products
Mathematical minima
Perceptron convergence procedure
Polynomials
Rational functions
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ISSN0036-1429
DOI10.1137/0719077

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Summary: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 $\sigma \geqq (2 - \sqrt 3)^2$ is obtained.
ISSN:0036-1429
DOI:10.1137/0719077