Rational approximation of $x^n$
arXiv:1801.010927 citationsh-index: 67
AI Analysis
Provides a theoretical result for rational approximation theory, but is incremental as it extends known asymptotic formulas to a new function class.
The paper derives the asymptotic minimax error for rational approximation of x^n on [0,1], showing it matches the known formula for e^x on (-∞,0] with Halphen's constant, independent of n.
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 $(-\infty,0\kern .3pt]$.