Domains of uniform convergence of real rational chebyshev approximants
Abstract Let the function ƒ be real-valued and continuous on [-1,1]. Denote by Rn,n, n = 1,2, … the rational Chebyshev approximants of order (n,n) of f on the interval. The present paper is inspired by a conjecture raised up by A. A. Gonchar, namely: If all (starting with a number n0) rational functions Rn,n are analytic in some Joukowski’s ellipse Ɛ, then ƒ admits an analytic continuation from [-1,1] into Ɛ. We show that the hypothesis is true if the set of the free poles of Rn,n and the set of the alternation points are asymptotically “well distributed”.