Rational approximation of 𝐱ⁿ

Nakatsukasa, Yuji; Trefethen, Lloyd

doi:10.1090/proc/14187

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 Halphen’s constant. This is the same formula as for minimax approximation of $e^x$ on $(-\infty ,0]$.

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