Abstract
We investigate the behavior of zeros of best uniform polynomial approximants to a function f, which is continuous in a compact set E ⊂ ℂ and analytic on E̊, but not on E. Our results are related to a recent theorem of Blatt, Saff, and Simkani which roughly states that the zeros of a subsequence of best polynomial approximants distribute like the equilibrium measure for E. In contrast, we show that there might be another subsequence with zeros essentially all tending to ∞. Also, we investigate near best approximants. For rational best approximants we prove that its zeros and poles cannot all stay outside a neighborhood of E, unless f is analytic on E.
The research of this author was done while visiting the Institute for Constructive Mathematics, University of South Florida, Tampa.
The research of this author was supported, in part, by the National Science Foundation under grant DMS-8620098.
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© 1988 D. Reidel Publishing Company
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Grothmann, R., Saff, E.B. (1988). On the Behavior of Zeros and Poles of Best Uniform Polynomial and Rational Approximants. In: Cuyt, A. (eds) Nonlinear Numerical Methods and Rational Approximation. Mathematics and Its Applications, vol 43. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2901-2_3
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DOI: https://doi.org/10.1007/978-94-009-2901-2_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7807-8
Online ISBN: 978-94-009-2901-2
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