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Ray sequences of best rational approximants for ${|x|}^{\alpha }$. (English) Zbl 0893.41011
Let ${r}_{mn}$ be the best rational approximation, in the sup norm, with numerator degree $m$ and the denominator degree $n$ to the function ${|x|}^{\alpha }$, $\alpha >0$, on $\left[-1,1\right]$. The authors examine the convergence behavior of so-called ray sequences $\left\{{r}_{mn}\right\}$, i.e. the sequences where $m$ and $n$ increase with the asymptotic ratio $\frac{m}{n}\approx c\ge 1$. The asymptotic distribution of the extreme points is analysed. There are the points of $\left[-1,1\right]$ where the error attains its sup norm. The optimal estimate of the sup norm of the error under consideration was recently obtained by one of the authors (Stahl) for the function $|x|$ and also for ${|x|}^{\alpha }$. The extension of this results to the estimation of the lower bounds is discussed.
MSC:
 41A25 Rate of convergence, degree of approximation 41A44 Best constants (approximations and expansions)
Keywords:
rate of convergence; best estimates