Asymptotic distribution of poles and zeros of best rational approximants to $x^α$ on [0,1]

E. Saff; H. Stahl

ArticleOriginal scientific text

Title

Asymptotic distribution of poles and zeros of best rational approximants to $x^{α}$ on [0,1]

Authors ¹, ²

Affiliations

Institute for Constructive Mathematics, Department of Mathematics, University of South Florida, Tampa, Florida 33620, U.S.A.
TFH-Berlin/FB2, Luxemburgerstr. 10, 13353 Berlin 65, Germany

Abstract

Let

r_{n} \cdot \in ℛ_{\cap}

be the best rational approximant to

f (x) = x^{α}

, 1 > α > 0, on [0,1] in the uniform norm. It is well known that all poles and zeros of

r_{n} \cdot

lie on the negative axis

ℝ_{< 0}

. In the present paper we investigate the asymptotic distribution of these poles and zeros as n → ∞. In addition we determine the asymptotic distribution of the extreme points of the error function

e_{n} = f - r_{n} \cdot

on [0,1], and survey related convergence results.

Keywords

ENG

rational approximation, best approximation, distribution of poles and zeros

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Title

Asymptotic distribution of poles and zeros of best rational approximants to αxα on [0,1]

Affiliations

Abstract

Keywords

Bibliography

Asymptotic distribution of poles and zeros of best rational approximants to $x^{α}$ on [0,1]