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

Volume 31 / 1995

E. Saff, H. Stahl Banach Center Publications 31 (1995), 329-348 DOI: 10.4064/-31-1-329-348


Let $r_n* ∈ ℛ_{nn}$ 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 $r_n*$ 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 $e_n = f - r_n*$ on [0,1], and survey related convergence results.


  • E. Saff
  • H. Stahl

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