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Behavior of zeros of polynomials of near best approximation. (English) Zbl 0605.41026
The purpose of this paper is to study the asymptotic behaviour of the zeros of polynomials of near best approximation to continuous functions f on a compact set E in the case when f is analytic on the interior of E but not everywhere on the boundary. For example suppose E is a finite union of compact intervals of the real line and f is a continuous function on E, but is not analytic on E, then the authors show that every point of E is a limit point of zeros of the polynomials of best uniform approximation to f on E. This fact answers a question posed by P. B. Borwein [Approximation theory IV, Proc. int. Conf., Tex. A & M Univ. 1983, 365-368 (1983; Zbl 0555.41021)].
Reviewer: S.M.Mazhar

MSC:
41A50Best approximation, Chebyshev systems
30C15Zeros of polynomials, etc. (one complex variable)
30E10Approximation in the complex domain