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Poles and zeros of best rational approximants of $|x|$. (English) Zbl 0815.41015

Let ${r}_{n}^{*}=p/q$ (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 $|x|$ on $\left[-1,1\right]$ and ${E}_{n,n}={E}_{n,n}\left(|x|,\left[-1,1\right]\right)$ – 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: ${lim}_{n\to \infty }{e}^{\pi \sqrt{n}}{E}_{n,n}\left(|x|,\left[-1,1\right]\right)=8$. The present paper completes the above one by an exhaustive analysis of an asymptotic behaviour of poles and zeros of ${r}_{n}^{*}$ and of the extreme points of error function $|x|-{r}_{n}^{*}$. More precisely, the author gives, for the large $n$, the formulas are individual location of the above points. Let ${r}_{n}^{*}=p/q$ (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 $|x|$ on $\left[-1,1\right]$ and ${E}_{n,n}={E}_{n,n}\left(|x|,\left[-1·1\right]\right)$ – 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: ${lim}_{n\to \infty }{e}^{\pi \sqrt{n}}{E}_{n,n}\left(|x|,\left[-1,1\right]\right)=8$. The present paper completes the above one by an exhaustive analysis of an asymptotic behaviour of poles and zeros of $r{*}_{n}$ and of the extreme points of error function $|x|-{r}_{n}^{*}$. More precisely, the author gives, for the large $n$, the formulas for individual location of the above points.

##### MSC:
 41A20 Approximation by rational functions 41A25 Rate of convergence, degree of approximation 41A44 Best constants (approximations and expansions)
##### Keywords:
rational approximation; asymptotic behaviour
##### References:
 [1] [BIS]H.-P. Blatt, A. Iserles, E. B. Saff (1987):Remarks on the behavior of zeros and poles of best approximating polynomials and rational functions. In: Algorithms for Approximation (J. C. Mason, M. G. Cox, eds.). Institute of Mathematics and Its Applications Conference Series, vol. 10. Oxford: Clarendon Press, pp. 437-445. [2] [BS]H.-P. Blatt, E. B. Saff (1986):Behavior of zeros of polynomials of near best approximation. J. Approx. Theory,46:323-344. · Zbl 0605.41026 · doi:10.1016/0021-9045(86)90069-9 [3] [Bu]A. P. Bulanov (1968):Asymptotics for the least deviation of |x| from rational functions. Mat. Sb.,76(118):288-303 (English translation (1968): Math USSR-SB,5:275-290). [4] [Ga]T. Ganelius (1979):Rational approximation of x ? on [0, 1]. Anal. Math.,5:19-33. · Zbl 0425.41017 · doi:10.1007/BF02079347 [5] [Go]G. M. Golusin (1957): Geometrische Funktionentheorie. Berlin: Deutscher Verlag der Wissenschaften. [6] [GR]I. S. Gradshteyn, I. M. Ryshik (1980): Table of Integrals, Series, and Products. New York: Academic Press. [7] [HK]W. K. Hayman, P. B. Kennedy (1976): Subharmonic Functions. London: Academic Press. [8] [La]N. S. Landkof (1972): Foundations of Modern Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 190. New York: Springer-Verlag. [9] [Me]G. Meinardus (1967): Approximation of Functions: Theory and Numerical Methods. New York: Springer-Verlag. [10] [Ne]D. J. Newman (1964):Rational approximation to |x|. Michigan Math. J.,11:11-14. · Zbl 0138.04402 · doi:10.1307/mmj/1028999029 [11] [St1]H. Stahl (1992):Best uniform rational approximation of |x| on [?1, 1]. Mat. Sb.,183:85-118. [12] [St2]H. Stahl (1992):Uniform approximation of |x| In: Methods of Approximation Theory in Complex Analysis and Mathematical Physics, (A. A. Gonchar, E. B. Saff, eds.). Moscow: Nauka, pp. 110-130. [13] [Va]R. S. Varga (1990): Scientific Computation on Mathematical Problems and Conjectures. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 60. Philadelphia: CBMS. [14] [Vj1]N. S. Vjacheslavov (1975):On the uniform approximation of |x| by rational functions. Dokl. Adad. Nauk SSSR,220:512-515 (English translation 1975: Soviet Math. Dokl.,16: 100-104). [15] [Vj2]N. S. Vjacheslavov (1980):On the approximation of x ? by rational functions. Izv. Akad. Nauk SSSR Ser. Mat.,44:92-109 (English translation (1981): Math. USSR. Izv.,16:83-101). [16] [VRC]R. S. Varga, A. Ruttan, A. J. Carpenter (1991):Numerical results on best uniform rational approximation of |x| on [?1,1]. Mat. Sb.,182(11):1523-1541 (English translation (1992): Math. USSR-Sb.,74(2):271-290).