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Rational approximation of $ \mathbf{x}^n$

Authors: Yuji Nakatsukasa and Lloyd N. Trefethen
Journal: Proc. Amer. Math. Soc. 146 (2018), 5219-5224
MSC (2010): Primary 41A20
Published electronically: September 4, 2018
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Abstract: Let $ E_{kk}^{(n)}$ denote the minimax (i.e., best supremum norm) error in approximation of $ x^n$ on $ [\kern .3pt 0,1]$ by rational functions of type $ (k,k)$ with $ k<n$. We show that in an appropriate limit $ E_{kk}^{(n)} \sim 2\kern .3pt H^{k+1/2}$ independently of $ n$, where $ H \approx 1/9.28903$ is Halphen's constant. This is the same formula as for minimax approximation of $ e^x$ on $ (-\infty ,0\kern .3pt]$.

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Additional Information

Yuji Nakatsukasa
Affiliation: Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom
Address at time of publication: National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan

Lloyd N. Trefethen
Affiliation: Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom

Received by editor(s): January 3, 2018
Received by editor(s) in revised form: March 26, 2018
Published electronically: September 4, 2018
Communicated by: Yuan Xu
Article copyright: © Copyright 2018 American Mathematical Society

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