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Rational Interpolation of the Exponential Function

Published online by Cambridge University Press:  20 November 2018

L. Baratchart ,
E. B. Saff  and
F. Wielonsky
L. Baratchart
Affiliation:
INRIA 2004, Route des Lucioles B.P. 93 06902 Sophia Antipolis Cedex France e-mail: [email protected]
E. B. Saff
Affiliation:
Department of Mathematics University of South Florida Tampa, Florida 33620 U.S.A. e-mail: [email protected]
F. Wielonsky
Affiliation:
INRIA 2004, Route des Lucioles B.P. 93 06902 Sophia Antipolis Cedex France e-mail: [email protected]
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Abstract

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Let m, n be nonnegative integers and B(m+n) be a set of m + n + 1 real interpolation points (not necessarily distinct). Let Rm,n = Pm,n/Qm.n be the unique rational function with deg Pm,n ≤ m, deg Qm,n ≤ n, that interpolates ex in the points of B(m+n). If m = mv, n = nv with mv + nv → ∞, and mv / nv → λ as v → ∞, and the sets B(m+n) are uniformly bounded, we show that locally uniformly in the complex plane C, where the normalization Qm,n(0) = 1 has been imposed. Moreover, for any compact set K ⊂ C we obtain sharp estimates for the error |ez — Rm,n(z)| when z ∈ K. These results generalize properties of the classical Padé approximants. Our convergence theorems also apply to best (real) Lp rational approximants to ex on a finite real interval.

Keywords

41A0541A21
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 6 , 01 December 1995 , pp. 1121 - 1147
Copyright
Copyright © Canadian Mathematical Society 1995

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