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Zbl 0815.41015
Stahl, H.
Poles and zeros of best rational approximants of $\vert x\vert$.
(English)
[J] Constructive Approximation 10, No.4, 469-522 (1994). ISSN 0176-4276; ISSN 1432-0940/e

Let $r\sp*\sb n= p/q$ (where $p$ and $q$ are polynomials of degree at most $n$, with real coefficients) be a best rational approximation in the Chebyshev sense (sup-norm) of $\vert x\vert$ on $[-1,1]$ and $E\sb{n,n}= E\sb{n,n} (\vert x\vert, [-1,1])$ -- the minimal approximation error. \par In a previous paper the author has proved an important result, conjectured early by Varga et al., namely the strong asymptotic error formula: $\lim\sb{n\to\infty} e\sp{\pi \sqrt{n}} E\sb{n,n} (\vert x\vert, [-1,1])=8$. The present paper completes the above one by an exhaustive analysis of an asymptotic behaviour of poles and zeros of $r\sp*\sb n$ and of the extreme points of error function $\vert x\vert- r\sp*\sb n$. More precisely, the author gives, for the large $n$, the formulas are individual location of the above points. Let $r\sp*\sb n= p/q$ (where $p$ and $q$ are polynomials of degree at most $n$, with real coefficients) be a best rational approximation in the Chebyshev sense (sup-norm) of $\vert x\vert$ on $[-1,1]$ and $E\sb{n,n}= E\sb{n,n} (\vert x\vert, [-1.1])$ -- the minimal approximation error. In a previous paper the author has proved an important result, conjectured early by Varga et al., namely the strong asymptotic error formula: $\lim\sb{n\to \infty} e\sp{\pi \sqrt{n}} E\sb{n,n} (\vert x\vert, [-1,1])= 8$. The present paper completes the above one by an exhaustive analysis of an asymptotic behaviour of poles and zeros of $r*\sb n$ and of the extreme points of error function $\vert x\vert- r\sp*\sb n$. More precisely, the author gives, for the large $n$, the formulas for individual location of the above points.
[J.Gilewicz (Marseille)]
MSC 2000:
*41A20 Approximation by rational functions
41A25 Degree of approximation, etc.
41A44 Best constants

Keywords: rational approximation; asymptotic behaviour

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