Poles and zeros of best rational approximants of |x|

The asymptotic distribution (forn→∞) of poles and zeros of best rational approximantsr _n ^* ∈R _nn of the function |x| on ?1, 1] as well as the asymptotic distribution of extreme points of the error function |x|?r _n ^* (x) on ?1, 1] is investigated. The precision of the asymptotic formulae corresponds to that of the strong error formula $\lim _{n \to \infty } e^{\pi \sqrt n } E_{nn} (|x|, - 1,1]) = 8$ , which has been proved in St1]. Here,E _nn (|x|, ?1, 1]) denotes the minimal approximation error in the uniform norm on ?1, 1]. The accuracy of the asymptotic distribution functions is so high that the location of individual poles, zeros, and extreme points can be distinguished forn sufficiently large.