Abstract view
Ray sequences of best rational approximants for $x^\alpha$


Published:19971001
Printed: Oct 1997
Abstract
The convergence behavior of best uniform rational
approximations $r^\ast_{mn}$ with numerator degree~$m$
and denominator degree~$n$ to the function $x^\alpha$,
$\alpha>0$, on $[1,1]$ is investigated. It is assumed
that the indices $(m,n)$ progress along a ray sequence in
the lower triangle of the Walsh table, {\it i.e.} the
sequence of indices $\{ (m,n)\}$ satisfies
$$
{m\over n}\rightarrow c\in [1, \infty)\quad\hbox{as } m+
n\rightarrow\infty.
$$
In addition to the convergence behavior, the asymptotic
distribution of poles and zeros of the approximants and the
distribution of the extreme points of the error function
$x^\alpha  r^\ast_{mn} (x)$ on $[1,1]$ will be studied.
The results will be compared with those for paradiagonal
sequences $(m=n+2[\alpha/2])$ and for sequences of best
polynomial approximants.