Rational approximation of $\mathbf {x}^n$
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- by Yuji Nakatsukasa and Lloyd N. Trefethen PDF
- Proc. Amer. Math. Soc. 146 (2018), 5219-5224 Request permission
Abstract:
Let $E_{kk}^{(n)}$ denote the minimax (i.e., best supremum norm) error in approximation of $x^n$ on $[0,1]$ by rational functions of type $(k,k)$ with $k<n$. We show that in an appropriate limit $E_{kk}^{(n)} \sim 2 H^{k+1/2}$ independently of $n$, where $H \approx 1/9.28903$ is Halphenâs constant. This is the same formula as for minimax approximation of $e^x$ on $(-\infty ,0]$.References
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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
- MR Author ID: 887438
- Email: nakatsukasa@nii.ac.jp
- Lloyd N. Trefethen
- Affiliation: Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom
- MR Author ID: 174135
- Email: trefethen@maths.ox.ac.uk
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5219-5224
- MSC (2010): Primary 41A20
- DOI: https://doi.org/10.1090/proc/14187
- MathSciNet review: 3866860