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# Ray sequences of best rational approximants for $|x|^\alpha$

Published:1997-10-01
Printed: Oct 1997
• E. B. Saff
• H. Stahl
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## 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.
 Keywords: Walsh table, rational approximation, best approximation, distribution of poles and zeros.
 MSC Classifications: 41A25 - Rate of convergence, degree of approximation 41A44 - Best constants

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